时间序列平稳性检验及ARIMA模型的应用

1、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。 数据来源：http:年份季度GDP （亿元）年份季度GDP （亿元）2000121329.92009174053.1224043.4283981.3325712.5390014.1429194.34101032.82001124086.42010187616.7226726.6299532.4328333.33106238.7431716.84119642.5200212629520111104641.3229194.82119174.333

2、1257.33126981.6434970.34138503.32003129825.520121117593.9232537.32131682.5335291.93138622.2439767.44152468.92004134544.6201311297472005238700.8201421439673418553152905.3446739.84168625.1140453.31140618.3244793.12156461.3348047.83165711.9454024.84181182.52006147078.920151150986.7252673.32168503356064

3、.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0 Series GDP Workfilr UN TITLEDiUntrtledV- B xvieijprocobjeclproperti«5P PrimNameFreBae"saripje <jenrLsheel Graph 5tais产rAugmented Dic

4、kty>Full*r Unit Root Teal on GDPNl Hypothesis: GEP has a unit rootIzKogenQus' DonslartLength. 5 ；ALtu nalic -ba匈 on SIC. tridxlay-lO)t-iitatisticProb,Auarncnicd Dickcv Fuller test Giotstc1 6530420 0997Test cntral ybIum1 % level-3 5401965% level2 9092001。牌 Icve2 582215l/acKinncr (19) one-sided

5、 p-v3lues.图1-1.GDP单位根检验结果（有截距项无趋势项）0 Series GDP Wbrkfile UNTITlFD:Untrtled- X门卜0寸 Frcperie< Print | Name- Fie ezeF Sample Ge nr Sh pet I Grap hi StatsTldeAugmented Dickey Fuller UnSt Root Test on GDPNull Hypcthesis: GDP has a unrt ractExDflenous: Constant Lrear TrendLag Lerti. 5 (AutomMc - based

6、on SIC. maxlag=W)t-SjatisticProb*lamented Dickcy Fuictcoi staiistc234 "100027Tea critical va ues: 1 % level.4.1130175% level3*18397。10% lev国-3 171071MxKinron1 田况)sne-sided p-values图1-2.GDP单位根检验结果（有截距项有趋势项）Senes <5DP W/orkiiip: UNTlTl ED：UntitlRd-exProciebjedPopeilies.Primr Jar- hRmzb5dtnplig

7、'GenrshtetGraptiM.心Augmented Dick«y-Fuller Unrt Root Te«t on GDPMull -1yco：lesis: GUP has a jnt raoiExogenous- NoneLmg Length 5 lALitcmatic - based ort SlQ rriaKlag=lO)t-SldiilicProb-Auamented Dicl(ev Fuller lest ststistc22026740.9929Test criiical values1 1% level 5% level 10% IcvdZ602

8、7Q41 06161-1 613396粗1前 Krunn (19黑)nnp-sided p-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I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。对原序列进行一阶差分再做单位根检验，结果如下:Augmented Dickey-Fuller Unit Root Test on DfGDPjNull Mpothcsim D(GDP-i hos a unit rootExojjerous NoneLag Length: 4 (Autorra

9、tic - based on SC, maxlag=10)t-Slatistic ProbAuqmcntcd 口inks，Fdlortcst st"i3配Q.O%3E4O.£侬Test critical values1 % level-2 3027945% level-1,W16110% level1.613398MacKinnon (19峋 口resided p-values图1-4.GDP 一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:0 SeriK- GDP Wcrkfile: Uhl TITLED ：Umi

10、tled- n.械wproc。均Propert/ | Pnnti Name Freeze Samp Ie Genrjsheti Gm ph StatsAugmented Dlck»y-Full»r Unit Reef T&st on D(GDP,2)Null Hypothesis: D(GDP,2) has a unit rootExogenous McneLdg Lesiglh 3 (Aulomdli' * Lased on SIC, rriaKldy-10>t-StatisticProb-Ajcmcntcd Die Fuller test stat

11、ic4 佝 52"UOOQOTest eritfcai valuer1 levelS' level-"&416110% level1 613388Tt-1ac Kj n non (19&6 ore-3 ice d p-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*kgrim of 3GDP,2)Date: 01 D7118 Time： 12:5占”中恒2C-MQ1 261阮14Inc udec ot servaiiori. .6A

12、utdcorrelatium Partial 匚 ur侔以ioiiAC FAC Q Suit Prob11 -0 aS 库8 3d3 L' OL J1匚2 a 33S -01&2 霞 &32 0 0)a1113 -0 6*3 -0 第4 E". 3g I - 01)11110 1A。.目峭 D.og 122 07 0.0001I1i6 o tip? 0178 H&13 a 0001二1 16 O3O2MSQ /5 95 0 00911? 0 fiSfi 0DO3 t77 S8 nOfH)11 15 77B -0.030 224 03 口。叩II1i9

13、0 527 -0 ME 71 0 则1I10 0.270 -0 004 252 53 0.0001=1l111 -0 466 0Q2A 270 50 OOM|1l |12 0 672 J,：J42 3U 03 0,0001I13 -0 455 -D018 325 55 0 0001o114 U 238 口.gT筑 045 000911i15 -0410 4)010 35 25 0 000H1i1< 吃 J J10 3 旺 66 L C. J11i117 -0Ja4 0&19 M5 0G 0 0W11118 0 20t. -0 033/8 0 0lH>=1l19 -0J4C

14、0 027 405 36 0.003IIl20 0 4懈 0 002 4加方 DOW二11 :21 H34I -0 056 小10 80 口叩。1l1 122 0 177 -0 0S3 4* 00 0 OOJ匚1123 - 293 -0.M2 452 99 0.00Jhm124 0 420 -0 041 47"4 OOM口11-2& -0 268 0 027 4S0.91 0.0(K»1 1126 0 147 0tt18 463 33 0 0W匚1127。第5 0002 ISO 23 0 0(K>11=112B 0 354 0 003 5G4 99 0 000

15、匚1129 -Q243 -0.0Q9 512 1& Q UW11加 0 123。必 SI4 06 0 ODO|_11J >31 0.104! O.J70 61D.08 O.QDJ1二122 0 2B9 0J42 5M0S OOM11133 0 205 0 010 5&&«2 口 0001口 1134 0 100 -0.037 537 22 0 000I-1113S 0 If® -0 (K3 自口安 DOMR11 3 235 -a.063 3013 0.000)1137 0 1EJ6 -0 003 563 40 0 0(K>111况 a.O

16、ZS 0 014 554.40 U.OUOid11 139 0 12S 0029 557 15 DOW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)。回I EquabCM EQ02 '/jrXfile; O/AJnUtledX- = «|vww|pr&c Ob|«cl Orintj Namt Freus j£rt>rwltFo<<castStats|R*5id5|OeperMferH Vnr iijbk- D

17、 (0 GDPj )>/e(nod. Lea st Squares口和 0I/07/1S ime 1307Sample (adjusted)： WWQQ4 2016Q4Included oteervaian ”针臼 adjusimemsConvergenEe ac hieued after 3 teiratic nsVari tsb IeCocffkicflftStd Error t-StafusocMX) 6706890093130-7 200503。叩胤)R-squared0 447 例Mean dependent var2切 5SOOAdjusted H-squdi trd0.44

18、7430S.D. dependefit var19附工00S r offtgrcssion14715 06Akaike info erfeerion22 M5MSum equaled 心就1.39E + 10Schwarz 口 it® ionZ2.”烟L og ReMnod-715 5079HaniiAriQuinr HDurtHn-Watson stat2.254710Invelec Ayi-GT图2.2 GDP二阶差分AR(1)模型回归结果=| E里jauort UNTITLED WorkTile： Q7：Untitlecl廿EProt_b tetprintNacre电Lnima

19、tEF-or&caaDependeE Var后加 DD(GDP)t/ethod- Least Squares口ate-01/0订 18 Time- 13:10Sample (adjust) 2001Q1 201 弓Q4Inr luded nbservHl 匕n 三 fi4 sfter djumeTitsConvergers e achieved after 2 re ratio nsVariableCoefficientStd Errrt-StatisdcProfc.AR0 3419550.1189412 374gBs0 0055R-sqjared0.11581BMean deoend

20、enl '/sr26&S437Adjusted R-qudrecI0115S1B£ D cepen Jent var13951 15S E of regression10760 2)8Akaike i a erttenor22 53237Sum squared resid222E+10Schwara cnt&non22/61。Log 冰白fMKd-720.0359Hannan-unn enter.22.54566Durbri-Watson 型 at2 S8827sInvcrlcd AH Roots58*50图2.3 GDP二阶差分AR(2)模型回归结果r亘 Equstiw JNTTTLED Workfile； O7：Untitled_ n xviewProc j Object一，Fr I