计量经济分析（第六版）答案 finalfall00VIP

EconometricsI.TakeHomeFinalExam.TodayisThursdayDecember14.ThisexamisduebyFriday,December22.Youmaysubmityouranswerstomeelectronicallyasanattachmenttoane-mailifyouwish.Therearefivepartsworth20%each.Recall,thisexamprovides40%ofyourgradeforthiscourse.Thefollowing25observationsareusedforthispartoftheexamination:16.00014.0008.00016.00014.0008.00022.0008.000014.00023.00022.00015.00021.00015.00025.00016.000Read;Nobs=25;Nvar=l;Names=Y;ByVariables$23.00025.00017.00023.00014.00022.00010.00018.00014.00016.00019.00027.000(ALIMDEPREADcommandisincludedifyouwishtouseit.YoucanjusttransplantthisintotheeditorinLIMDEPandexecuteittoinputthedata.)SupposewebelievethatthedataonYaregeneratedbyapoissondistribution.Then,theprobabilitydensityfunctionforYisf(Y)=exp(-X)XY/Y!LetX=exp(a)Wearegoingtoestimatetheparameteroc.TOC\o"1-5"\h\zPOISSON;Lhs=Y;Rhs=ONE$+ +PoissonRegression |MaximumLikelihoodEstimates |Dependentvariable Y |Weightingvariable ONE INumberofobservations 25 |Iterationscompleted 5 |Loglikelihoodfunction -78.30529 |Chi-squared=37.50783RsqP=.0000|G-squared= 39.52722RsqD= .0000 |Overdispersiontests:g=mu(i) : 1.447 |Overdispersiontests:g=mu(i)人2: 1.447 | 十+ + + + + + +IVariable | Coefficient | StandardError |b/St.Er.|P[|Z|>z] | Meanof X|+ + + + + + +Constant2.883682770 .47298377E-01 60.968 .0000Thetablegivestheestimateofa.Whatistheestimatedasymptoticdistribution?Theexpectedvalueoftherandomvariable,Yis口二九=exp(a).Estimate日usingyourmaximumlikelihoodestimate.Estimatetheasymptoticstandarderrorofthisestimator.Presenta95%confidenceintervalfortheparameterbasedonyourresults.Since二E[Y]isX,youshouldbeabletoestimatewiththesamplemeanoftheobservationsonY.Doso,anddescribeyourfinding.Usingthefamiliarformulaforthevarianceofthemean,estimatethestandarderrorofthisestimator,andcompareyourresulttothatin(b).Thevarianceofthisrandomvariableisa2=X.Youshouldbeabletoestimatec2withthesamplevarianceoftheobservationsonY.Doso,andcompareyourestimatetotheoneyougetbyusingtheMLEinthetable.Doesthedifferenceappeartobesmallorisitlargeenoughtomakeyoususpectthatthemodelwhichhasthesamemeanandvarianceisincorrect?Howmightyoutestthisassumption?

ILContinuingpartI,wealsohavethefollowingdataonXRead;Nobs=25;Nvar=l;Names=X;ByVariables$16111223232122241215222116201715171417231915252212Wewillnowformulateakindofregressionmodel.WebelievethatY|XhasthePoissondistributionspecifiedearlier,butnow,X=Exp[a+px]Thetablebelowpresentsthemaximumlikelihoodestimatesoftheparametersofthismodel.+ IPoissonRegressionIMaximumLikelihoodEstimatesYONE255-68.12529-78.3052920.36001YONE255-68.12529-78.3052920.360011.6414944E-05IWeightingvariableINumberofobservationsIIterationscompletedILoglikelihoodfunctionIRestrictedloglikelihoodIChi-squaredIDegreesoffreedomISignificancelevelIChi-squared=IG-squared=IOverdispersionIOverdispersion+ 18.7309319.16722IChi-squared=IG-squared=IOverdispersionIOverdispersion+ 18.7309319.16722RsqP=RsqD=.5006.5151tests:g=mu(i) :-1.360tests:g=mu(i)A2:-1.591TOC\o"1-5"\h\z+ + + + + + +IVariable | Coefficient | StandardError |b/St.Er.|P[|Z|>z] | Meanof X|+ + + + + + +Constant1.924770881 .22497947 8.555 .0000X .5152813743E-01 .11545125E-01 4.463 .0000 18.160000Asbefore,thisisXwhichisnowWeareinterestedintheexpectedvalueofY|X.Asbefore,thisisXwhichisnowE[Y|X]=exp(a+pX)Usingyourresultsabove,estimatetheslopeofthisregressionatthemeanofX(18.16).LinearlyregressYonaconstantandX.Whatistheslopeinthisregression.Comparethisslopetothemaximumlikelihoodestimates.Theproceduresin(a)and(b)abovesuggesttwomethodsofestimatingaandp.Comparethetwointermsofconsistencyandefficiency.SinceE[Y|X]isafairlysimplefunctionofX,youmightalsoconsidernonlinearleastsquaresestimationofaandp.Describeindetailhowtocomputethenonlinearleastsquaresestimatesofaandp.Howwouldyoucomputeasymptoticstandarderrorsforyourestimators?(e)HowwouldyouformaconfidenceintervalforyourestimateofE[Y|X=X].UsingtheresultsinpartsIandII,testthehypothesisthatPequals0usingaWaldtestandusingalikelihoodratiotest.DescribehowonewouldcarryoutaLagrangemultipliertestofthishypothesis.Thefollowingquestionsarebasedontheregressionmodel:Y=01+p2*X+(33*Z+04*XZ+p5*D+££isassumedtobezeromean,homoscedastic,andnonautocorrelated.Thefollowingdataareobtained:(notethatXZistheproduct,XtimesZ.)YXZXZD6.544956.185792.7446216.9776.0000005.019148.203002.9578824.26351.0000020.2805.9287391.648391.53092.00000015.77133.671902.346338.615491.0000015.32443.200562.796358.94989.0000007.274129.499232.0856719.81231.00000-2.327039.743622.7390926.6887.00000013.00438.572271.8325715.70931.0000012.377214.49951.4521421.05531.000001.876549.157492.6600324.3592.0000006.059849.914961.9052018.8900.00000013.28948.802481.088609.58238.00000018.86155.255471.555138.172941.0000016.66771.514291.569882.37725.00000021.08265.439691.073805.84114.000000-11.994113.77182.8295738.9683.00000018.47801.798222.819295.06970.0000001.3483611.36362.5403028.86701.000009.7277811.53761.8909621.81711.0000021.37924.682371.348366.313521.0000016.32217.201461.372089.880981.0000021.56793.536082.241737.926941.000004.751339.288012.2102220.5285.00000010.06324.797552.2640510.8619.00000015.417913.42511.1515415.4595.000000Estimatetheparametersofthemodelusingordinaryleastsquares.Presentallresultsandexplainyourcomputations.Inadditiontotheslopes,estimatetheparametero,thestandarddeviationof8.TestthehypothesisthatneitherXnorZhaveanyexplanatorypowerintermsofexplainingvariationinY.TestthehypothesisthatZdoesnothaveanyexplanatorypowerinexplainingvariationinY.TestthehypothesisthatthecoefficientsonXandZintheregressionareequal.Dothistestintwoways:Useonlythestatisticalresultsoffittingthefullregression.Fittheregressionwiththerestrictionimposed,andtestthehypothesisusingtheresultsofbothregressions.(Note,ignorethevariableXZinthiscomputation.)6.WeareinterestedinexaminingthemarginaleffectofchangesinXonE[Y|X,Z,D].Whatis5E[y|X,Z,D]/6X?ComputethiseffectwithZequaltoitsmean.Howwouldyoucomputeastandarderrorfortheestimateofthiseffect?Howwouldyoutestthehypothesisthatthiseffectequalszero?V.Thedatalistedabovearenowassumedtocomefromaprocessinwhichthereisalinearregressionmodel,butpossiblyaheteroscedasticdisturbance.TheregressionequationisY=pi+02*X+03*X+04*XZ+05*D+8shasmean0,butmaybeheteroscedastic.EstimationinthispartoftheexamisbasedonthedatayouusedinpartIV.Supposethatthetruevarianceof8isVar[e]=c2*Exp(X*D)Ifyouestimatethebetasusingordinaryleastsquares,whatarethepropertiesoftheestimator?(Bias,consistency,efficiency,truecovariancematrix.)Supposeyoubelievethatthevarianceof8isa2Exp(X*D),but,infact,thetruevarianceisjustcy2.(I.e.,yourbeliefismistaken.)SupposeyoufitthemodelbyGLSinspiteofthetruevariance.Whatarethepropertiesofyourestimator?(Note,youcanusetrueGLShere,sincetherearenofreeparametersinthevariancefunction.)Computethetwoestimatorsyoudescribedinparts1and2,andreportallresults.(Note,inpart2,therearenoparametersinthevariancepart,soyoucancomputethetrueGLSestimator.)ComparethevariancesoftheOLSandGLSestimator,bothtrueandestimated.Usingtheleastsquaresresults,computetheWhiteestimatorforthevarianceoftheOLSestimator.Describewhyyouwoulddothiscomputation.Supposethetruemodelis,infactVarfs]=W*Exp(aXD)whereaisaparametertobeestimated.Howwouldyoutestthehypothesisthatalphaequals1.0againstthealternativehypothesisthatalphaisnotequalto1.0?Givefull