NYUNYU中STERNDepartmentofEconomicsNEWYORKUNIVERSITYLEONARDN.STERNSCHOOLOFBUSINESSECONOMETRICSIFall2007,TR,l：00-2：20ProfessorWilliamGreenePhone:212.998.0876Office:KMC7-78 Homepage:/^wgreeneEmail:wgreene@URLforcoursewebpage:/^wgreene/Econometrics/Econometrics.htmAssignment6AsymptoticsforLeastSquaresandtheClassicalRegressionModelThisexerciseisbasedonthe'AirlinesData'usedinafewexamplesinyourtext.Themodelinthisproblemsetislogc二a+piogq+y(log2q/2)+8ilog(pf)+521f+53t+e.(SeeTableF7.1onpage949.Thedataareonthewebsiteforthetext.)Linearregressionoflog(cost)=1conaconstant(one),log(output)=Iq,onehalfthesquareofIq=.5*lq2=lq2,thelogofthefuelprice,Ipf,andtheloadfactor,If,andatimetrend,t,producethefollowingresults(onthenextpage).Notethattherearethreesetsofresultsshown.TestthehypothesisthatboththeloadfactorandtimetrendcoefficientsarezerousinganFtest.Notethatthisrequiresvaluesfrombothregressionsbelow.Now,usingonlythefirstsetofresults,useaWald(chi-squared)statistic,testthesamehypothesis.Explainthedifferenceinthetwotests.Whatistheresultofthetest?F二((1.577479-1.149658)/2)/[1.149658/(90-6)]=15.63.Large.Rejecthypothesis.TheWaldstatisticiscomputedbelow.Forthelinearmodel,Wald=2*F.Ingeneral,thechi-squaredisthelimitingstatisticwhenthedegreesoffreedomgrowslarge.

-->regr;lhs=lc;rhs=onezIq,lq2,Ipf,If,t$+-1Ordinaryleastsquaresregression11ModelwasestimatedDec09,2005at04:30:11PM |1LHS=LCMean =13.36561 |1Standarddeviation=1.131971 |1WTS=noneNumberofobservs. =90 |1ModelsizeParameters =6 |1Degreesoffreedom=84 |1ResidualsSumofsquares =1.149658 |1Standarderrorofe=.1169889 |1FitR-squared =.9899189 |1AdjustedR-squared=.9893188 |1ModeltestF[5, 84](prob)=1649.68(.0000)|1DiagnosticLoglikelihood =68.51107 |1Restricted(b=0) =-138.3581 |1Chi-sq[ 5](prob)=413.74(.0000)|1Infocriter.LogAmemiyaPrd.Crt.=-4.226814 |1AkaikeInfo.Criter.=-4.227012 |1AutocorrelDurbin-WatsonStat. =.2169784 |1Rho=cor[e,e(-1)] =.8915108 |+- ++ + + 4- + 4- +IVariable| Coefficient | StandardError |t-ratio |P[|T|>t] | Meanof X|+ + + + + + +Constant11.3580713.6771772416.773.0000LQ.93040362.0280246533.199.0000-1.17430919LQ2.04602849.021442532.147.03471.34409282LPF.28511398.060163614.739.000012.7703594LF-1.56214282.32709516-4.776.0000*56046016T.03409548.011270133.025.00338.00000000-->matr;r=[0,0,0,0,1,0/0,0,0,0,0,1];aq=[0/0]$-->matr;list;m=r*b-aq;vm=R*varb*r*;Wald=m*<vm>m$MatrixM has2rowsand1columns.1+ 1| -1.562142| .03410MatrixVM has2rowsand2columns.1 2+ 1| .10699 9202913D-042|-.9202913D-04 .00013MatrixWALDhas1rowsand1columns.1+ 1| 31.25883-->calc;list;fc=wald/2$FC= .15629415347651660D+02Thethirdsetofregressionresultsgivencorrespondstoamodellogc=a+plogq+Silf+甑+￡Theestimateofthe'scale'parameterinthismodelisE=1/p.Usingthethirdsetofregressionresults,estimatethisquantityandcomputeanestimateoftheasymptoticstandarderror.Whatistheasymptoticdistributionofthisestimator?-->Regress;lhs=lc;rhs=one,lqzIf,t$+ IOrdinaryIModelwasILHS=LCleastsquaresregressionestimatedDec09,2005at04:34:43PMWTS=noneModelsizeResidualsFitMeanStandarddeviationNumberofobservs.ParametersDegreesoffreedomSumofsquaresStandarderrorofeR-squaredAdjustedR-squared13.365611.131971+ IOrdinaryIModelwasILHS=LCleastsquaresregressionestimatedDec09,2005at04:34:43PMWTS=noneModelsizeResidualsFitMeanStandarddeviationNumberofobservs.ParametersDegreesoffreedomSumofsquaresStandarderrorofeR-squaredAdjustedR-squared13.365611.131971904861.515892.1327654.9867075.9862438Modeltest

DiagnosticF[3,86](prob)=2127.93(.0000)Infocriter.AutocorrelLoglikelihoodRestricted(b=0)Chi-sq[ 3](prob)LogAmemiyaPrd.Crt.AkaikeInfo.Criter.Durbin-WatsonStat.Rho=cor[e,e(-1)]56.06677-138.3581388.85(.0000)-3.994858-3.994917.3755908.81220464-4-4-4-IVariable|Coefficient|StandardError|t-ratio|P[|T|>t]|MeanofX|Constant14.5121911.1970700773.640.0000LQ.86736709.0141300961.384.0000LF-1.42812721.36398158-3.924.0002T.08404797.0040523620.741.0000+++++-1.17430919.560460168.00000000->calc;list;beta=b(2);e=1/beta$BETA =.86736708567543600D+00E =.11529143963553570D+01Calculator:Computed2scalarresults->calc;list;vbeta=varb(2,2);dedb=-l/betaA2$VBETA =.19965951595452470D-03DEDB =-.13292116053234360D+01Calculator:Computed2scalarresults->calc;list;ve=dedbA2*vbeta;sde=sqr(ve)$VE =.35275912994487820D-03SDE =.18781883024470100D-01TheestimateofEis1.153.Theestimatedasymptoticstandarderroris.01878.Theasymptoticdistributionwouldbenormalwiththismeanandstandarddeviation.c.OnecommonmeasureofeconomiesofscaleinstatisticalcostanalysisisthemeasureE=l/[61ogc/aiogq]=- ，+/logqUsingtheresultsfromthefirstequation,estimateEcomputedattheaveragevalueoflogq(notthelogoftheaverageofq).Computetheasymptoticstandarderrorforyourestimate.Explainyourcalculationsateachstep.+ +1Ordinaryleastsquaresregression1IModelwasestimatedDec09,2005at04:45:35PM |ILHS=LCMean=13.36561 |1Standarddeviation=1.131971 |IWTS=noneNumberofobservs.=90 |IModelsizeParameters=6 |1Degreesoffreedom=84 |1ResidualsSumofsquares=1.149658 |1Standarderrorofe=.1169889 |IFitR-squared= .9899189 |AdjustedR-squared= .9893188 |1ModeltestF[5, 84](prob)=1649.68(.0000)|IDiagnosticLoglikelihood= 68.51107 |Restricted(b=0)=-138.3581 |Chi-sq[ 5](prob)=413.74(.0000)|1Infocriter.LogAmemiyaPrd?Crt.=-4.226814 |AkaikeInfo.Criter.=-4.227012 |IAutocorrelDurbin-WatsonStat.= .2169784 |Rho=cor[e,e(-1)]= .8915108 |++ + + + + + IVariable | Coefficient| StandardError|t-ratio |P[|T|>t] | MeanofX+ +- + + -+ ——十 Constant11.3580713.6771772416.773.0000LQ.93040362.0280246533.199.0000-1.17430919LQ2.04602849.021442532.147.03471.34409282LPF.28511398.060163614.739.000012.7703594LF-1.56214282.32709516-4.776.0000.56046016T.03409548.011270133.025.00338.00000000->calc;beta=b(2);gamma=b(3)$->calc;logqb=xbr(Iq)$>matr;vbc=varb(2:3,2:3)$->calc;list;e=1/(beta+gamma*logqb)$E = .11410940699966370D+01一->calc;list;dedb=-e人2;dedg=-eA2*logqb$DEDB = 13020956765814900D+01DEDG = .15290629193330670D+01Calculator:Computed2scalarresults->matr;list;ge=[dedb,dedg]$MatrixGE has1rowsand2columns.1 24- 1| -1.30210 1.52906->matr;;ve2=ge*vbc*ge1$-->calc;list;sqr(ve2)$Result= .16416912272909100D-01Theestimateis1.141.Theestimatedasymptoticstandarderorris.01642.2.Continuingpartcofthepreviousexercise,thelogquadraticcostfunctionimpliesa'U'shapedaveragecostfunction.Thatinturnimpliesthatthereareeconomiesofscaleinairlineservices,butthattheyrunoutatsomepoint,sothattheadvantageoflargescaleeventuallydisappears.Inparticular,Ididthefollowing:(1)Computethemeansoflog(pf),If,andt,(2)usingtheregressionmodelestimatedcoefficients,computeestimatedcostsforalargerangeofvaluesofq(3)plotfittedcostagainstq.Thefigureappearsbelow

1.95001.65001.87501.80001.7250EstinatodAvoiageCostFunctbn1.95001.65001.87501.80001.7250EstinatodAvoiageCostFunctbnAsyoucansee,thecostcurvehasabottom,somewherenear4.Thisiscalledtheefficientscale.Theefficientscaleoccurswheretherearenoeconomiesofscale,thatis,wherethescaleelasticityinquestionl.b.equals1.Theinterestingquantityistheqwherethisoccurs,whichforthiscostfunctionwouldbeUsingtheregressionresultsforthefirst(long)regressionbelow,estimateq*andestimatetheasymptoticstandarderrorforyourestimator.Formaconfidenceintervalforq*?Whatistheasymptoticdistribution.->calc;list;escale=exp((1-beta)/gamma)$ESCALE= .45359223882484530D+01->calc;desdb=-escale/gamma;desdg=-escale*(1-beta)/gamma/X2$->matr;list;ges=[desdb,desdg]$MatrixGEShas1rowsand2columns.1 2+ 1| -98.54598 -149.00432->matr;;ves=ges*vbc*ges1$->calc;list;ses=sqr(ves)$SES= .57962303216844540D+01->calc;list;lower=escale-1.96*ses;upper=escale+1.96*ses$LOWER = 68246890422530770D+01UPPER =?15896533818749980D+02

TOC\o"1-5"\h\zIOrdinaryleastsquaresregression |I LHS=LC Mean = 13.36561 |I Standarddeviation = 1.131971 |I WTS=none Numberofobservs. = 90 |I Modelsize Parameters = 6 |I Degreesoffreedom = 84 |I Residuals Sumofsquares = 1.149658 II Standarderrorofe = .1169889 |I Fit R-squared = .9899189 |I AdjustedR-squared = .9893188 |IModeltestF[5, 84](prob)=1649.68(.0000)|+ ++ + + + + + +IVariable | Coefficient | StandardError |t-ratio |P[|T|>t] | Meanof X|+ + + + + + +Constant11.3580713.6771772416.773.0000LQ.93040362.0280246533.199.0000-1.17430919LQ2.04602849.021442532.147.03471.34409282LPF.28511398.060163614.739.000012.7703594LF-1.56214282.32709516-4.776.0000*56046016T.03409548.011270133.025.00338.00000000EstimatedAsymptoticcovariancematrixfortheestimatedcoefficients:0.458569 -0.00111089 -0.000782679 -0.0393469 -0.0248664 0.00721773-0.00111089 0.000785381 0.000536681 0.000142168 -0.000631672 -1.8712e-005-0.000782679 0.000536681 0.000459782 1.98541e-005 0.000915807 3.51257e-006-0.0393469 0.000142168 1.98541e-005 0.00361966 -0.00284521 -0.000642815-0.0248664 -0.000631672 0.000915807 -0.00284521 0.106991 -9.20291e-0050.00721773 -1.8712e-005 3.51257e-006-0.000642815 -9.20291e-005 0.000127016TOC\o"1-5"\h\z+ +IOrdinaryleastsquaresregression |IModelwasestimatedOct08,2003at01:23:58PM |I LHS=LC Mean = 13.36561 |I Standarddeviation = 1.131971 |I WTS=none Numberofobservs. = 90 |I Modelsize Parameters = 4 |I Degreesoffreedom = 86 |I Residuals Sumofsquares = 1.577479 |I Standarderrorofe = .1354355 |I Fit R-squared = .9861674 |I AdjustedR-squared = .9856849 |IModeltestF[3, 86](prob)=2043.74(.0000)|+ ++ + + + + + 4-|Variable | Coefficient | StandardError |t-ratio |P[|T|>t] I Meanof X|EstimatedAsymptoticcovariancematrixfor

1 2+ + + + + + +EstimatedAsymptoticcovariancematrixfor

1 2Constant9.1382275