A Toolkit for Analyzing Nonlinear Dynamic Stochast

DiscussionPaper101FederalReserveBankofMinneapolisInstituteforEmpiricalMacroeconomics250MarquetteAvenueMinneapolis,Minnesota55480-0291June1995AToolkitforAnalyzingNonlinearDynamicStochasticModelsEasilyHaraldUhligCentER,UniversityofTilburgABSTRACTOften,researcherswishtoanalyzenonlineardynamicdiscrete-timestochasticmodels.Thispaperprovidesatoolkitforsolvingsuchmodelseasily,buildingonlog-linearizingthenecessaryequa-tionscharacterizingtheequilibriumandsolvingfortherecursiveequilibriumlawofmotionwiththemethodofundeterminedcoecients.Theinnovationhereistodemonstratethatlog-linearizingthenonlinearequationscanusuallybedonewithouttheneedforexplicitdierentiation,toextendthemethodofundeterminedcoecientstomodelswithmoreendogenousstatevariablesthanex-pectationalequations,toprovideageneralsolutionandtoprovidefrequency-domaintechniquestocalculatethesecondorderpropertiesofthemodelinitsHP-lteredversionwithoutresortingtosimulations.SincethemethodisanEuler-equationbasedapproachratherthananapproachbasedonsolvingasocialplannersproblem,modelswithexternalitiesordistortionarytaxationdonotposeadditionalproblems.MATLABprogramstocarryoutthecalculationsinthispa-peraremadeavailable.ThispapershouldbeusefulforresearchersandPh.D.studentsalike.Correspondingaddress:CentERforEconomicResearch,TilburgUniversity,Postbus90153,5000LETilburg,TheNetherlands,e-mail:[e-mail],JanMagnusandEllenMcGrattanforhelpfulcomments.ThispaperwascompletedwhilevisitingtheInstituteforEmpiricalMacroeconomicsattheFederalReserveBankofMinneapolis:Iamgratefulforitshospitality.AnyviewsexpressedherearethoseoftheauthorsandnotnecessarilythoseoftheFederalReserveBankofMinneapolisortheFederalReserveSystem.11IntroductionOften,researcherswishtoanalyzenonlineardynamicdiscrete-timestochasticmodels.Thispaperprovidesatoolkitforsolvingsuchmodelseasily,buildingonlog-linearizingthenecessaryequationscharacterizingtheequilibriumandsolvingfortherecursiveequilibriumlawofmotionwiththemethodofundeterminedcoecients.WhilethesemethodshavebeenpioneeredbyMcCallum(1983),King,PlosserandRebelo(1987)aswellasCampbell(1994),thenewcontributionhereistodemonstratethatlog-linearizingthenonlinearequationscanusuallybedonewithouttheneedforexplicitdierentiation,toextendthemethodofundeterminedcoecientstomodelswithmoreendogenousstatevariablesthanexpectationalequations,toprovideageneralsolutionandtoprovidefrequency-domaintechniques,buildingonresultsinKingandRebelo(1993),tocalculatethesecond-ordermomentsofthemodelinitsHP-lteredversionwithoutresortingtosimulations.SincethemethodisanEuler-equationbasedapproachratherthananapproachbasedonsolvingasocialplannersproblem,solvingmodelswithexternalitiesordistortionarytaxationdoesnotposeadditionalproblems.Sincethe(nonlinear)Eulerequationsusuallyneedtobecalculatedinanycaseinordertondthesteadystate,applyingthemethoddescribedinthispaperrequireslittleintermsofadditionalmanipulationsbyhand,givensomepreprogrammedroutinestocarryoutthematrixcalculationsofsection5.MATLABprogramstocarryoutthesecalculations,giventhelog-linearizedsystem,areavailableattheFederalReserveBankofMinneapoliswebsite1.Themethodinthispaperthereforeallowstosolvenonlineardynamicstochasticmodelseasily.Numericalsolutionmethodsforsolvingnonlinearstochasticdynamicmodelshavebeenstudiedextensivelyintheliterature,seeinparticularKydlandandPrescott(1982),thecomparisonbyTaylorandUhlig(1990)andthemethodsproposedbyvariousau-thorsinthesameissue,Judd(1991),HansenandPrescott(1995)andDanthineandDonaldson(1995).Theliteratureonsolvinglinear-quadraticdynamicstochasticmodelsisevenlarger,seee.g.thetextbooktreatmentinSargent(1987),ChaptersIXandXI,aswellas,say,BlanchardandKahn(1980),McGrattan(1994)orHansen,McGrattanandSargent(1994).Subjecttoapplicability,allthemethodrelyingonaloglinearapproximationtothesteadystatehaveincommonthattheywillndthesamerecursiveequilibriumlawofmotionasthemethoddescribedinthispaper,sincethelinearspaceapproximatinganonlineardierentiablefunctionisuniqueand\immunetodierentiabletransformationsoftheparameterspace.ButwhileMc-Grattan(1994)andHansen,McGrattanandSargent(1994)focusonsolvingmodels1Theaddressofthatwebsiteis(1980)solvelinearsystemsbysearchingforthestablemanifoldintheentiresystemofneces-saryequationsdescribingtheequilibriumrelationships,thispaperbycontrastsolvesdirectlyforthedesiredrecursiveequilibriumlawofmotion.Thisapproachisverynatural.Thestabilityconditionisimposedatthepoint,whereacertainmatrixquadraticequationissolved.Itisshownhowthismatrixquadraticequationcanbereducedtoastandardeigenvalueproblemofanothermatrixwithtwiceasmanydimensions.TworelatedcontributionsareMcCallum(1983),whichisthekeyreferenceforthemethodofundeterminedcoecients,andCeriaandRios-Rull(1992).Whilethesetwocontrib