recursive macroeconomic theory time series

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1、Chapter2TimeSeries2.1.TwoworkhorsesThischapterdescribestwotractablemodelsoftimeseries:Markovchainsandfirst-orderstochasticlineardifferenceequations.Thesemodelsareorganizingdevicesthatputrestrictionsonasequenceofrandomvectors.Theyareusefulbecausetheydescribeatimeserieswithparsimony.Inlaterchapters

2、,weshallmaketwouseseachofMarkovchainsandstochasticlineardifferenceequations:(1)torepresenttheexogenousinformationflowsimpingingonanagentoraneconomy,and(2)torepresentanoptimumorequilibriumoutcomeofagents’decisionmaking.TheMarkovchainandthefirst-orderstochasticlineardiffer-encebothuseasharpnotionofas

3、tatevector.Astatevectorsummarizestheinformationaboutthecurrentpositionofasystemthatisrelevantfordetermin-ingitsfuture.TheMarkovchainandthestochasticlineardifferenceequationwillbeusefultoolsforstudyingdynamicoptimizationproblems.2.2.MarkovchainsAstochasticprocessisasequenceofrandomvectors.Forus,t

4、hesequencewillbeorderedbyatimeindex,takentobetheintegersinthisbook.Sowestudydiscretetimemodels.Westudyadiscrete-statestochasticprocesswiththefollowingproperty:MarkovProperty:Astochasticprocess{xt}issaidtohavetheMarkovpropertyifforallk≥1andallt,Prob(xt+1

5、xt,xt−1,...,xt−k)=Prob(xt+1

6、xt).Weassumet

7、heMarkovpropertyandcharacterizetheprocessbyaMarkovchain.Atime-invariantMarkovchainisdefinedbyatripleofobjects,namely,–29–30TimeSeriesann-dimensionalstatespaceconsistingofvectorsei,i=1,...,n,whereeiisann×1unitvectorwhoseithentryis1andallotherentriesarezero;ann×ntransitionmatrixP,whichrecordsthepr

8、obabilitiesofmovingfromonevalueofthestatetoanotherinoneperiod;andan(n×1)vectorπ0whoseithelementistheprobabilityofbeinginstateiattime0:π0i=Prob(x0=ei).TheelementsofmatrixParePij=Prob(xt+1=ej

9、xt=ei).Fortheseinterpretationstobevalid,thematrixPandthevectorπ0mustsatisfythefollowingassumption:Assumpt

10、ionM:a.Fori=1,...,n,thematrixPsatisfiesnPij=1.(2.2.1)j=1b.Thevectorπ0satisfiesnπ0i=1.i=1AmatrixPthatsatisfiesproperty(2.2.1)iscalledastochasticmatrix.Astochasticmatrixdefinestheprobabilitiesofmovingfromonevalueofthestatetoanotherino