Stochastic simulation

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1、StochasticsimulationswithDYNARE.Apracticalguide.FabriceCollard(GREMAQ,UniversityofToulouse)AdaptedforDynare2.5.2byMichelJuillard(CEPREMAPandUniversityParis8)Firstdraft:February2001Thisdraft:January2003.Thisdocumentdescribesamodelinvolvingbothendogenousandexogenousstatevariable.Wefirstdescr

2、ibethetheoreticalmodel,beforeshowinghowtheperturbationmethodisimplementedinDYNARE(MATLABversion).1AtheoreticalmodelWeconsideraneconomythatconsistsofalargenumberofdynastichouseholdsandalargenumberoffirms.Firmsareproducingahomogeneousfinalproductthatcanbeeitherconsumedorinvestedbymeansofcapit

3、alandlaborservices.Firmsowntheircapitalstockandhirelaborsuppliedbythehouseholds.Householdsownthefirms.Ineachandeveryperiodthreeperfectlycompetitivemarketsopen—themarketsforconsumptiongoods,laborservices,andfinancialcapitalintheformoffirms’shares.Householdpreferencesarecharacterizedbythelifet

4、imeutilityfunction:!X∞1+ψ?τ−thtEtβlog(ct)−θ(1)1+ψτ=twhere0<β?<1isaconstantdiscountfactor,ctisconsumptioninperiodt,htisthefractionoftotalavailabletimedevotedtoproductiveactivityinperiodt,θ>0andψ>0.Weassumethatthereexistsacentralplannerthatdetermineshours,consumptionandcapitalaccumulationma

5、ximizingthehousehold’sutilityfunctionsubjecttothefollowingbudgetconstraintct+it=yt(2)whereitisinvestmentandytisoutput.Investmentisusedtoformphysicalcapital,whichaccumulatesinthestandardformas:kt+1=exp(bt)it+(1−δ)ktwith0<δ<1(3)whereδistheconstantphysicaldepreciationrate.btisashockaffecting

6、incorpo-ratedtechnologicalprogress,whichpropertieswillbedefinedlater.1Outputisproducedbymeansofcapitalandlaborservices,relyingonaconstantreturnstoscaletechnologyrepresentedbythefollowingCobb–Douglasproductionfunction:y=exp(a)kαh1−αwith0<α<1(4)ttttatrepresentsastochasticshocktotechnologyorS

7、olowresidual.Weassumethattheshockstotechnologyaredistributedwithzeromean,butdisplaybothpersis-tenceacrosstimeandcorrelationinthecurrentperiod.Letusconsiderthejointprocess(at,bt)definedasatρτat−1εt=+(5)btτρbt−1νtwhere

8、ρ+τ

9、<1and

10、ρ−τ

11、<1forsakeofstationarityandE(