管理学院考博资料——计量经济学Lecture 7.ppt

Econometrics(I)Lecture7AutocorrelationandRegressionwithTimeSeriesData,主讲人：,2,SeriallyCorrelatedDisturbances,Fortimeseriesregressionmodel,3,SourcesofSerialCorrelation,Inertiainmosteconomictimeseries:GDP,Priceindexes,etcSpecificationbiasOmittedvariablesIncorrectfunctionalformsLaggeddependentvariablesDatatransformation:Quarterlydata,firstdifferencing,etcNonstationarity:Mean,variance,andcovarianceofatimeseriesdonotchangeovertime,4,TheLagOperator,LYt=Yt-1,L2Yt=L(LYt)=Yt-2LqYt=Yt-q,L0=I,L0Yt=YtThelagoperatorcanbetreatedasascalarPolynomials,5,First-orderAutocorrelation:AR(1),SpecificationToensurethatAR(1)isastationarystochasticprocess,Properties,6,First-orderAutocorrelation:AR(1),AR(1)suggeststhatuthasalongmemory,butwhens,thetimeinterval,goestoinfinity,theautocorrelationcoefficientconvergestozero.Var-CovmatrixAR(p),7,First-orderMovingAverage:MA(1),SpecificationProperties,8,First-orderMovingAverage:MA(1),MA(1)suggeststhatuthasarathershortmemory:TheerrorsareonlyinterrelatedintwosuccessiveperiodsVar-CovmatrixMA(q),9,RelationsbetweenARandMA,AR(1)andMA()MA(1)andAR(),10,ConsequencesofAutocorrelation,TheOLSestimatorsarestilllinear,unbiased,consistent,andasymptoticallynormalHowever,theyarenolongerefficientNormallyunderestimatingtheerrorvarianceUnderestimatingandinflatingt-statisticsTheproblemsarenotresolvedbyusinglargesamplesizes,11,DetectingAutocorrelation,Informal/GraphicalMethodsDurbin-WatsonTestAssumptions:ThedisturbancesaregeneratedbyAR(1)processandarenormallydistributed.Also,theregressionmodeldoesnotincludethelaggedvaluesofthedependentvariableasoneoftheexplanatoryvariablesH0:Teststatistic,12,DetectingAutocorrelation,Durbin-WatsonTestTheexactprobabilitydistributionofD.W.isunknown,sounliket,Ftests,thereisnouniquecriticalvaluethatwillleadtotherejectionofH0However,wecanderivealowerbounddLandanupperbounddUdependingonTandk(Alreadytabulated)DecisionrulesDisadvantages,13,DetectingAutocorrelation,TheBreusch-GodfreyTestItisageneraltestoftheAR(p)andMA(q)processesLaggeddependentvariablescanbeincludedProceduresEstimatetheoriginalmodelbyOLStoobtaintheresidualsRegresstheresidualsontheoriginalXsandObtainR2fromthisauxiliaryregressionItcanbeshownthat,sorejectthenullhypothesisofnoautocorrelationiftheteststatisticissufficientlylargeNotethatthisisessentiallyaLagrangeMultipliertest,14,CorrectingOLSStandardErrors,HAC(heteroscedasticity-andautocorrelaton-consistent)standarderrorsaredevelopedbyNeweyandWest(1987)Itisanextensionofheteroskedasticity-robuststandarderrors(White,1980)Theyareonlyvalidinlargesamples,andwecangettheestimatorsofNoneedtoknowthestructureofautocorrelationandeasytoexecuteviastandardeconometricpackages,15,GLSEstimation,InthecaseofAR(1),16,GLSEstimation,Afterthetransformation(quasi-differencing)Fortheestimationtobefeasible,ThenitbecomesFGLSbypluggingtheestimatedAR(1)coefficientintothetransformedregressionmodel.TheFGLSestimatorsarebiasedbutconsistent,17,FGLSEstimation,Cochrane-Orcuttestimationomitsthefirstobservation,whereasPrais-Winstenestimationkeepsthefirstone.Itmakesnodifferenceinlargesamples,butsincethesizeofmanytimeseriessamplesissmall,thedifferencesarenotableintheseapplications.Iterativescheme:WhentheFGLSestimatorsarefoundusing,anewsetofresidualscanbecomputed,andaestimatorwouldbeeasytoobtain,thusleadingtothenewsetFGLSestimators.Wecanrepeattheprocessmanytimes,untiltheestimateofchangesverylittlefromthelastiteration.,18,AntidumpingFilingsandChemicalImports,TheeffectsofantidumpingfilingsbyU.S.chemicalindustriesonChineseimportsWereimportsunusuallyhighintheperiodimmediatelyprecedingtheinitialfiling?befile6=1duringthesixmonthsbeforefilingDidimportschangenoticeablyafteranantidumpingfiling?affile6=1duringthesixmonthsafterfilingWasthereasignificantreductioninChineseimportsafterthedecisioninfavoroftheU.S.industry?afdec6=1duringthesixmonthsafterthepositivedecisionDependentvariable:chnimp=volumeofChineseimportsControlvariableschempi=chemicalproductionindex(controlforoveralldemand)gas=volumeofgasolineproduction(controlfordemand)rtwex=exchangerateindex(strengthofdollaragainstothercurrencies)MonthlydatafromFebruary1978throughDecember1988,19,AntidumpingFilingsandChemicalImports,20,RegressionwithTimeSeriesData,Asequenceofrandomvariablesindexedbytimeiscalledastochasticprocessoratimeseriesprocess.Normally,weobtainoneandonlyonepossibleoutcome,orrealization,oftheprocess.Finitedistributedlag(FDL)modelsImpactpropensity/multiplier:ImmediatechangeinYduetoaone-unittemporaryincreaseinXattimetLong-runpropensity/multiplier:Long-runchangeinYgivenapermanentone-unitincreaseinX,21,OLSAssumptionsandProperties,LinearityandnoperfectcollinearityZeroconditionalmeanStrictexogeneity:Theerrorattimetisuncorrelatedwitheachexplanatoryvariablesineverytimeperiod.Incross-sectionaldata,randomsamplingsuggeststhatuiisautomaticallyindependentoftheexplanatoryvariablesotherthani,butintimeseriesregression,randomsamplingisalmostneverappropriate.Contemporaneousexogeneity,22,OLSAssumptionsandProperties,Linearity,noperfectcollinearity,andstrictlyexogeneityleadtounbiasedOLSestimatorsintimeseriesregressionZeroconditionalmean:Whatcausestheviolationofthestrictexogeneityassumption?Changesintheerrortermtodaycancausefuturechangesintheexplanatoryvariable.Thatis,theremayexistfeedbackfromYonfuturevaluesofX.Whereas,strictlyexogenousexplanatoryvariablescannotreacttowhathashappenedtoYinthepast.Rainfallandoutputvs.Policeofficerspercapitaandmurderrate,23,OLSAssumptionsandProperties,Homoskedasticity(likelytobeviolatedintimeseriesdata)andNoSerialCorrelationTheclassicallinearmodelassumptionsfortimeseriesdataaremorerestrictivethanthoseforcross-sectionaldata:StrictexogeneityandnoautocorrelationIncross-sectionaldata,randomsamplingensuresthatuiandujareindependentforanytwoobservationsiandj.AlltheaboveassumptionsleadtotheBLUE.Plusthenormalityassumption,standardinference(t,Ftests)canbeundertakenintimeseriesdata.,24,EffectsofPersonalTaxExemptiononFertilityRates,Variablesofinterestgfr:Generalfertilityrate,theNo.ofchildrenborntovery1,000womenofchild-bearingagepe:AveragerealdollarvalueofthepersonaltaxexemptionControlvariablesww2=1duringtheyears1941through1945pill=1fromyear1963on,whenthebirthcontrolpillwasmadeavailableYearlydatafrom1913through1984,25,RegressionresultResultwithaFDLmodelMulticollinearitybetweenpet,pet-1andpet-2makesithardtoestimatetheindividualeffectHowever,thethreevariablesarejointlysignificant.TheestimatedLRP=0.101,butisitstatisticallysignificant?,EffectsofPersonalTaxExemptiononFertilityRates,26,EffectsofPersonalTaxExemptiononFertilityRates,ReparameterizationRunningthenewregressionyieldsEventhougharenotsignificant,theestimateofLRPisourprimaryinterest,27,TrendingVariablesinOLSRegression,Manytimeserieshaveatendencyofgrowingordecliningovertime.Nothingabouttrendingvariablesnecessarilyviolatestheclassicallinearmodelassumptions.However,inmanycases,twotimeseriesprocessesappeartobecorrelatedonlybecausetheyarebothtrendingovertimeforreasonsrelatedtootherunobservedfactors:SpuriousregressionproblemFormulation:,28,29,TrendingVariablesinOLSRegression,Addingatimetrendtosolvetheomittedvariableproblem:“Detrending”Housinginvestmentandprices(1947-82,U.S.)PuertoRicanEmploymentRateandU.S.minimumwageandGNP(1950-97),30,StationaryTimeSeries,StationarystochasticprocessForeverycollectionoftimeindices,thejointdistributionofisthesameasthejointdistributionofNorestrictiononhowxtandxt-1arerelatedtooneanother,butitrequiresthenatureofanycorrelationbetweenadjacenttermsbethesameacrossalltimeperiodsStationarityisanaspectoftheunderlyingstochasticprocess,soitisveryhardtodeterminewhetherasinglerealizationwasgeneratedbyastationaryprocessornot.However,aprocesswithatimetrendisclearlynonstationary,31,CovarianceStationaryProcess,Astochasticprocessxt:t=1,2,withafinitesecondmomentE(xt2)iscovariancestationaryifE(xt)isconstantVar(xt)isconstnatForanyt,s1,Cov(xt,xt+s)dependsonlyonsandnotontIfastationaryprocesshasafinitesecondmoment,itmustbecovariancestationary,buttheconverseiscertainlynottrue.Ifweallowtherelationshipbetweentwovariablestochangearbitrarilyineachtimeperiod,wecannothopetolearnmuchabouthowachangeinonevariableaffectstheotherifweonlyhaveasinglerealizationWeassumeacertainformofstationarityinthatOLSestimatorsdonotchangeovertime,32,WeaklyDependentTimeSeries,Acovariancestationarytimeseriesisweaklydependentifthecorrelationbetweenxtandxt+sgoestozero“sufficientquickly”ass.Itisalsocalledasymptoticallyuncorrelated.Examples:MA(1)andAR(1)Thecentrallimittheoremfortimeseriesdatarequiresstationarityandsomeformofweakdependence.Atrendingseries,thoughcertainlynonstationary,canbeweaklydependent.Aseriesthatisstationaryaboutitstrend,aswellasweaklydependent,iscalledatread-stationaryprocess.ItcanbeusedinOLSregressionprovidedthattimetrendisincludedinthemodel,33,AsymptoticPropertiesofOLSwithTimeSeriesData,Stationarityandweakdependence(xt,Yt);t=1,2,isstationaryandweaklydependent,sothatthelawoflargenumbersandthecentrallimittheoremcanbeappliedtosampleaveragesContemporaneousexogeneityAlongwiththeusualassumptionsoflinearityandnoperfectcollinearity,theOLSestimatorsareconsistent:,34,Examples,Staticmodel:Underweakdependence,SoitdoesnotruleoutcorrelationbetweenXsandpasterrortermsorYs,forexample,FinitedistributedlagmodelAR(1)modelAmodelwithalaggeddependentvariablecannotsatisfythestrictexogeneityassumptionHowever,theOLSestimatorsoftheAR(1)modelareconsistent,thoughbiased.Thebiascanbelargeifsamplesizeissmallorif1iscloseto1.,35,HACStandardErrorsandFGLSRevisited,FGLSestimatorsareconsistentifdataareweaklydependentandexplanatoryvariablesarestrictlyexogenous.(OLSjustrequiresweakdependencyandcontemporaneousexogeneity.)ItcanbeshownthattheweakestconditionfortheconsistencyofFGLSisthatutisuncorrelatedwithxt-1,xt,andxt+1HACstandarderrorscanbepoorlybehavedinthepresenceofsmallsamplesizeandsevereserialcorrelation(OLSestimatorsaretooinefficient)ComputingHACstandarderrorsafterFGLSestimationmaybeagoodapproach,36,HighlyPersistent(StronglyDependent)TimeSeries,RandomWalkItcanbeshownthatSoitisneitherstationarynorweaklydependentRandomwalkisaspecialcaseofwhatisknownasaunitrootprocessPolicyrelevance,37,38,HighlyPersistent(StronglyDependent)TimeSeries,Aseriescanbetrendingbutnothighlypersistent(trendstationary),andhighlypersistentseries(interestrates,inflationrates,unemploymentrates)havenoobvioustrends.Randomwalkwithadrift,39,40,HighlyPersistent(StronglyDependent)TimeSeries,Weaklydependentprocessaresaidtobeintegratedoforderzero,orI(0).Nothingneedstobedonetosuchseriesbeforeusingtheminregressionanalysis.Unitrootprocessesaresaidtobeintegratedoforderone,orI(1).