时间序列平稳性检验及ARIMA模型的应用

时间序列模型的应用以2000年第一季度-2016年第四季度中国国内生产总值统计数据为例，建立ARIMA模型。所有数据按当年价格计算，共有68个观测值。］年份季度GDP（亿元）年份季度GDP（亿元）2000121329.92009174053.1224043.4283981.3325712.5390014.1429194.34101032.82001124086.42010187616.7226726.6299532.4328333.33106238.7431716.84119642.5200212629520111104641.3229194.82119174.3331257.33126981.6434970.34138503.32003129825.520121117593.9232537.32131682.5335291.93138622.2439767.44152468.92004134544.620131129747I数据来源：/easyquery.htm?cn=B01

238700.821439673418553152905.3446739.84168625.12005140453.320141140618.3244793.12156461.3348047.83165711.9454024.84181182.52006147078.920151150986.7252673.32168503356064.73176710.4463621.64192851.9200715717720161161572.7264809.62180743.7369524.33190529.5478721.44211281.32008169410.4278769382541.9488794.31、利用ADF检验检查序列的平稳性

PTCobjEc?:TChKrri"PrintLManeFreessanpeGrm<5tnrSheet匚hEMtzder0SeriesGDPWorkfilrUNTITLEDilJntrtledV-BxAugm*nt*dDickeAFullerUnitRootT»slonGQPJJIHypothesis:GEPhasaunitrootIzKogenQus'DonslartLength.5；ALtuJJIHypothesis:GEPhasaunitrootIzKogenQus'DonslartLength.5；ALtunalic-bdMonSIC.tridxlay-lO)AStatisticProb1AuarncnicdDickcvFullertestGiotstc165304200997Testcntrfllvalueslevel-3E401985怙level■2909200I'D%tevel2582215l/acKinnor(1A9A)one-sidedp-vNues.图1-1.GDP单位根检验结果（有截距项无趋势项）0Seifes:GDPWorkhle:UNTinED::Unlitled\ViewLProc：］ObjectPrepertie*PnntN〃meFr?p?eSamplrGenirSheetGraphJit北IrkAugmentedDiektyFullerUnrtRootTestonGDPNulll-Aipcthesis:GDPhasaunrtractExDflenous:ConstantLrearTrendLagLerAti.5(AutomAic-basedonSIC.maxlagAW)t-SiatisticProb*AanenreddickcyFuiorteststaiistc2.34?71U0/1027Teacriticalvaues:1%levelb4.113017亏％levtl-3.A397010%官巾创-m171071AMxKinron(1S&5)sne-sidedp-values图1-2.GDP单位根检验结果（有截距项有趋势项）rh/j■e--es：COPVVrxkfirLNTlTlFD:l)ntitirAl\■□X"句叫ProcObjectProperties,PrimNameFr&ez@SampeG&nrShAetGraptifc.l=hdtr-♦AugmentedDick«y-FullerUnrtRootTestonGDPMull-lycoAesis:GUPhasajntraoiExogenous-NoneL自gLength5lALitcmatic-tasedortSlQrriaKlag=lO)t-Sldii—licProb.'AuamentedDicl<evFullerleststatistc2202674口.9929Testcriticalvalues:1%levelZ6027Q45%level10A6161-1613396*kgrKrunn(19胎*nnp-sidedp-v<ilirs图1-3.GDP单位根检验结果（无截距项无趋势项）以上三种形式检验结果中不能拒绝ADF检验原假设的概率分别为99.97%、40.27%、99.29%，故接受原假设，认为时间序列GDP是非平稳的。以上三种形式检验结果中不能拒绝对原序列进行一阶差分再做单位根检验，结果如下：AugmentedDicker-FullerUnitRootTestonD(GDP|NuiHypothesisD(GDP}hasaunirootExojjerousNoneLagLencth:4(Autorratic-basedonSIC,maxlag=10)[■StatisticProb*AuqmcntccDickeyF』|yto玄statistic0WG36406>1€6Testcriticalvalues1hlevel-23027945利levelA1.9A-16110%level1.613398MacKinnon(199气one-sidedp-values图1-4.GDP—阶差分单位根检验结果(无截距项无趋势项)可以看出一阶差分后，d(GDP)序列是不平稳的。阶差分结果如下：它Series：GDPWtorkfile：UhJTlTLE&：Untitled\-nVewProcObject3ropertiesPrntNdrrefreezeSairipkGenrSheetGraphStatsAugmentedDlck»y-Full»rUnitRootT&s(onD|GDP,2)NullHypothesis:D(GDP,2)hasaunitrootExogenousMarieLagLsigtt)3(AiMomAtx-basedonSIC,rr»udag=1D)t-StatisticProb■AjcmcntcdDickqyFullerteststatic斗A&277D0900Testurglvalues1%level5珂level」&4€16110%level■I613398Tt-1acKjnnon(19&6ore-3icedp-values.图1-5.GDP二阶差分单位根检验结果(无截距项无趋势项)因此，二阶差分平稳。2、确定模型形式检查GDP二阶差分序列的自相关函数和偏自相关函数:

Corr*kgrimofD(GDP^}Date:01D7'18Time：12:人5SdiTf)l£2C-MQ1201吧4IncudecotseAvaiiori.-.6AutocorrelationF*uaiCorrelationACFACQSuitPTcb1:1mqq056830773Q.05Q0688□1—1=□233旨□JU19236032oODaU130H忖-a884什、t的000011iW-0HIJ5JJr11)1HJ40人890.0@3012297o.ooa0■11617SHQ13O&i}1二1•460302-0.0301S595oooa111110w讣(KBfTTUSoon■1B□77B-0J30224aaoooa厂11i]ry讣-0031?«71000'}1□l1wa.2/o-00042b2.b30ODO1=1l11-04购0Q2427050OOQO||——l|1120J/2J.D423d[l」jO.OOQ1l13.0455.4)0183?555OOdQ1OI1140238o.oor33。45000011iII15-04104)01014525□ODOHi11435820.J103T£T00Q.QOO11i17』甜40.019测^DODO1.I1020641(X333»37600tf[>=1l15-0屈0.027405360000II■lfl2004EM01)0242921QOOO二11:21.03414)056地的oooa1□11.2?0177-0(a+14000000匚11123-0293-0.IM245299O.ODOhm1i24042C-00414M84000ftO112S4J.28G0027O.ODO1.11%0H7D0184&333OOQO11127000249023000011=111种0354oomfiG4船0ODO匚11I-□243-Q.0Q9&12旧DOOT111官巾ai23E1406DOMII11131-0.19All.U/EJSIL}080.0001二13202B90042530OSOQDOc1133020f.00106M92000ft1D|1340100-003753722oooaI-11.360If®0.1X13WO站00001n11[孙0235-0.063却13o.oao1111370166-000356340ODDO二1111比0.0790.014564.400_ODOid1139012S002955?1500(W1图2.1二阶差分自相关、偏自相关函数从上图可以看出自相关函数呈振荡型衰减(拖尾)，而偏自相关函数在3阶后截尾，可初步判断适合的模型为AR(1)、AR(2)、AR(3)。对GDP二阶差分分别做AR(1)、AR(2)、AR(3)模型分析，结果如下:M]Equahcjn,EQO±V/uMe：O7AJnUtled\-3K|vww|pr&cOb|«clOrintjNam#Freen]j£rt>rwlt[Fo<<cast[Statsaids|DeperMtemVariyfcteD(0(GDP)}>/e(nod.LeastSquares0眦！0I/07/1SAime1307Sample(adjusted)：200QQ42016Q4Incii.dedobsArvmticr*r.rmferatJ.usfnTentsCorib-ieTgenizeachievedaflwrteralionsVariableCocffkicflftStd[Zrrort-Stafusoc陌⑴-067M990093130-7200503OODDOR-squanedAdjustedR-sqiored合匚offogrcssianSumreAK：LoglihHihcndDurtHn-Wstsonstat04斗M3C亡etidependentvarS-0.A47430D-dtpendtifitvarAKDi«mfo1471506erterxinSchwarzcnteiion1.39E+10HaniiflriAQuinriLiz.71H祐792.2547102&35SOO1&?ftS.0O22046402207965rninveledARKost?r«7图2.2GDP二阶差分AR(1)模型回归结果叵EquationsUNTITLEDWo倾e:Q7：：Untitled\-3X-bitctprint电”w韭Forwaasta:s耳心fidMr>peiifirnlVaiiabt!D[D(GDP))t/ethod'LeastSquaresDateQ1；0TH8Time'13:10Sample(adjust)2001Q120伯Q4Inrhidednhseni'AlAnsfi4sfterAdjiiAmFTitsConvergerseachievedafter2rerationsVariableCoefficientStdEitmt-5utis：icProfc.AR⑵03419550.1189412e?A&as00055R-sq」ared0.115«诣Meandeoendenl尸尸26&S437AdjustedR-AqudreiJ0115S1BSDcepenJent1395115SEofregression1B7602GAkaikeirfnerttenor2253237Sumsquaredresid222E+10Schwaracnt&nonZ2.=CbWLoglAeliAood-720.0359Hannan-nnenter.22.54566Durbri-Watsonslat2606279IrA-crtcdAHRoot358*50卜■血ProcjObji&ct曲“FrmlNamx入freeze隹FarsoitStaUFZiEqusiion-JNTTTLED