unit-root nonstationary processes

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1、Tsay-Driver-12013/10/2813:34page265#1CHAPTER5Unit-RootNonstationaryProcessesInthepreviouschapters,wefocusedourstudyonweaklystationaryprocesseswhosefirsttwomomentsaretimeinvariant.Thereare,however,manyempiricaltimeseriesthatexhibittime-varyingproperties.Fo

2、rinstance,stockpricesandmacroeconomicvariablestendtoexhibitcertainstochastictrendssothattheirfirstmomentisoftentime-varying.Figure5.1showsthetimeplotofU.S.quarterlyrealgrossdomesticproduct(GDP),inlogarithm,fromthefirstquarterof1948tothesecondquarterof2012.Thep

3、lotshowsaclearupwardtrend.Inthischapter,westudysomespecialmultivariatetimeseriesthatexhibitcertainstochastictrends.Werefertosuchseriesasunit-rootnonstationaryprocesses.Thesimplestunit-rootnonstationarytimeseriesistheunivariaterandomwalk,whichcanbewrittenaszt

4、=zt−1+at,(5.1)where{at}isaniidsequenceofrandomvariateswithmeanzeroandfinitevarianceσ2.ThemodelinEquation(5.1)canbetreatedasanAR(1)modelwithφ=1,a1thatis,zt=φ1zt−1+at,φ1=1.(5.2)Thecharacteristicequationofthisspecialmodelis1−x=0,whichhasasolutionx=1.Consequently

5、,therandomwalkprocessztiscalledaunit-rootprocess.Thesolutionx=1isontheunitcirclesothatztisnotastationaryseries.Inpractice,onemightbeinterestedinverifyingthatanobservedtimeseriesztfollowsarandomwalkmodel.Tothisend,onemayconsiderthehypothesistesting:H0:φ1=1ver

6、susHa:φ1<1,whereφ1isthecoefficientofEquation(5.2)orinthefollowingequationzt=φ0+φ1zt−1+at.(5.3)MultivariateTimeSeriesAnalysis:WithRandFinancialApplications,FirstEdition.RueyS.Tsay.c2014JohnWiley&Sons,Inc.Published2014byJohnWiley&Sons,Inc.265Tsay-Driver-

7、12013/10/2813:34page266#2266unit-rootnonstationaryprocessesln(gdp)7.58.08.59.09.51950196019701980199020002010YearFIGURE5.1TimeplotofU.S.quarterlyrealGDP,inlogarithm,from1948.Ito2012.II.TheoriginalGDPwereinbillionsofchained2005dollarsandmeasuredto1decimal.T

8、histestingproblemiscalledunit-roottesting.Ithadattractedmuchattentioninthe1980s,andmanyteststatisticsareavailable.SeeTsay(2010,Chapter2),Hamilton(1994,Chapter15)andthereferencestherein.Weshallco