第九章ARIMA模型2范文

第九章ARIMA模型已知1867-1938年英国（英格兰及威尔士）绵羊的数量如表 1所示，运用时间序列模型预测未来三年英国的绵羊数量。表1 1867-1938 年英国绵羊数量年份绵羊数年份绵羊数年份绵羊数186722031891211119151752186823601892211919161795186922541893199119171717187021651894185919181648187120241895185619191512187220781896192419201338187322141897189219211383187422921898191619221344187522071899196819231384187621191900192819241484187721191901189819251597187821371902185019261686187921321903184119271707188019551904182419281640188117851905182319291611188217471906184319301632188318181907188019311775188419091908196819321850188519581909202919331809188618921910199619341653188719191911193319351648188818531912180519361665188918681913171319371627189019911914172619381791（—） 序列的平稳性检验（1）时序图在workfile的工作区，双击"X”，打开"Series:X”窗口，选择View/Graph,type下选择Basicgraph和Line&Symbol，点确定，得到图1。ruarrjFruarrjFuy」uu」p■■入」pyj12jpujju*』［j^jiij卜2』l、*1***118」［」puiupiij ”从图1可以看出，绵羊数量的序列 X具有向下的趋势，不平稳。对序列进行一阶差分，去除趋势。在命令行输入命令： genrDX=d(X)，回车，得到X差分后的序列D%画DX的时序图，见图2:图2从图2可以看出，差分后的序列已经没有趋势，可以初步判断是平稳的。序列DX平稳的单位根检验在Series:DX窗口，选择View/UnitRootTest ，弹出如下窗口(见图 3)。图3选择Testtype为AugmentedDickey-Fuller,Testforunitrootin 选择Level,Includeintestequation处依此选择Trendandintercept、intercept、None,Laglength处按默认的选项，由SchwarzInfoCriterion自动选择最佳滞后长度。在选择Trendandintercept 后，输出了表2.表2AugmentedDickey-FullerUnitRootTestonDXNullHypothesis:OXhasaunitrootExogenous:Constant,LinearTrendLagLength:1(AutomaticbasedonSIC,MAXLftG=11)t-StatisticProb*AugmentedDickev*Fullerteststatist!c-6.4035940.0000Testcriticalvalues:1%level5%level10%level-4.096614-3.476275-3165610*MacKlnnon(1996)one-sidedp-valuesAugmentedDickey-FullerTestEquationDependentvariable:D(DX)Method:LeastSquaresDate：05/05/10lime:22:10Sample(adjusted)：1E701938Includedobservations:69afteradjustmentsCoefficientStd.Errort-StatisticProb.DX(①-07995590.124861-6.4035840.0000D(DX(-1))0.3B10550.11151734170260.0011c-15.166701795742-08445930.4014@TREND(1967)0.3152100.42461707423390.4506R-squared0.387123Meandependentvar3.913043AdjustedR-squared0.35B83BS.D.deperidmntvarG7.37541SE.otregression699S336Akaikeinfocriterion1139006Sumsquaredresid318171,2Schwarzcriterion11.51957Loglikelihood-3B8.9570Hannan-Qjinncriter.11.44144F-statistic1168671Durbin-Watsonstat2.049490Prot)(F-statistic)0.000001根据表2的结果，ADF统计量=-6.，相应的P值为0.0000，小于〉=0.05，因此拒绝原假设（原假设为序列 DX有一个单位根，也即序列 DX非平稳），DX是平稳的。在表2中给出了单位根检验的辅助回归结果，其中 C和@trend（1867）均不显著（相应的t统计量的p值分别为0.4014,0.4606），因此检验模型中不应包括趋势和截距项。 在模型选择处（Ineludeintestequation）重新选择None，得到下表（表3）:表3AiigirneinedDtcKey-i-uiierunithooiIestonDXMullHypothesis:DXh朋aunitroot"xogenous:None_agLength:1(AutomaticbasedonSIC,MAXLAO=11)t-StatisticProb*AugmentedDickey-Fullertestslalistic-6.4306490.0000festcriticalvalues:1%level5%level10%level-2.538907-1945596-1,613719"MacKinnon(1996)one-sidedp-values.

ADF统计量=-6.，相应的P值为0.0000，小于：•=0.05，因此拒绝原假设，即 DX是平稳的。（4）DX的白噪声检验在Series:DX窗口，选择View/Correlogram，在CorrelogramSpecification下选择Level，点“OK'确定，得到图5。orrclogramSpecification XCorrelogramoF(<LevelQ丄或differenceO^nddifferenceLagstoinclude32CorrelogramofDXDate:O5JO5/1OTima：22:19Sample:18671936Includedobservations:71AutocorrelationPartialCorrelationACPACQ-StatProbI1]10.35103519.13820003ir1O12-0145-0306107080005i-1O13-0.409*029623.4350000二11[14-0.270-0.05529.0650.000I〕11J150.0720.10129.4680.000I□1|0160.163-008031.5900000I]1|[170076^0059320610000I11113-00300023321370000IL1119-00E2001332.4570000|[11110-0061-0.07632.7740000111111-0.0+0-0.04732.9160.001111C112-0.060-0.10133.4760.0011匸11口113-01110-0.12234.5480.001|L11□114-O.0B7-010935.2410.001II11B115-0012-006135.2540002II]1111GD0G1-001736.37200031n1n170.2740241 43.05400001]11口1130068-012543.9110001111>19-0.0+50.03S44.0120.001匚111120-0.200-0.06348.0790.000匚11□121-0.200-0.03052.2070.0001]>111220.0450034524130.0001□1111230136^005554414000011111240136-001456.4430.000图5从图5的最后一列，Q统计量的伴随概率P均小于：=0.05，因此应拒绝原假设（原假设为：序列为白噪声），即DX是非白噪声的。（二）模型的识别根据图5,可知DX序列的自相关系数4阶截尾，偏相关系数 3阶截尾，因此可选择 MA(4)、AR(3)、ARMA(3，4)模型拟合。相应地，X序列可选择ARIMA(0,1,4)、ARIMA(3,1,0)和ARIMA(3,1,4)。模型的估计及检验1、拟合ARIMA(0,1,4)模型命令：LSD(X)CMA(1)MA(2)MA(3)MA(4)表4DependentVariable:DQOMeth口d:LeastSquaresDate:05/05T10Time:22:34Sample(adjjsted):18681938Includedobservations：71alteradjustmentsConvergenceachievedafter24iterationsMABackcast18641867CoefTicientStdErrort*StatisticProb.c-7.1013572.184440-3.2511110.0018MA(1)03714970.1067333.48060700009MA⑵-02149010110342■1.94758500557MAO)-04565220.1Q7424■424972500001MA(4)-05960690114973-5.169660oooooR-squared0.389592Meandependentvar-5.002617AdjustedR-squared0.352593SD.dependentvar34.25550S.E,ofregression67.70307AkaIkeinfocriterion11.33962Sumsquaredrestd303329.4Schwarzcriterion11.49796Loglikelihood-3976209Hannari'Quirmcrlter.114019SF-statistic10.53111Curbin-Watsonstat1.998563Prob(F'statistic)0.000001InvertedMARoots.97-,23*.80i-.23+.80i-.89写出拟合的方程:(1—B)Xt=—7.1019(10.3715B—0.2149B2—0.4565B3—0.5967B4)；ts.e.=67.79307 ，AIC=11.33862，SC=11.49796模型的显著性检验：(在Equation窗口，选择View/residualtests/Correlogram-Q-Statictic:

CorrelogramofResidualsuaie.ujjuj/iuiinie.zz.jis~Sample:16681936Inch」clmdobseivations:71Q-statisticprobabilitiesadjustedfor4ARMAterm(s)ACFACQ-StatProbAutocorrelationPartialCorrelationACFACQ-StatProbiji* iji* □iiii]ii[iijiii | ii [ ii f it [ i)匚iini[ii【iir >iT|i1-0009-0.0090.00662011601161.01573-0.014-0.0131.031540.1120.100199965-0.054-0.0512.22690.136601280.1073.53310.1717-00030.0113.53360.31E80005-0.0323.53530.473g-0122-0.1144.77490.44410-0.076-0.1045.26910.51011*0.114*0.0070.49412-0.131-0.1377.9059044313-0.073-C.047S.38840.49614-0.077-0055@93090.53Q16-0093-0.0569.73360.56416-0124-0.08611.1760.514170137C.18012971U.45018-0106-0.061140770.444根据图6，残差序列的Q统计量的根据图6，残差序列的Q统计量的序列为白噪声)，即残差序列是白噪声，拟合模型显著有效。模型参数的显著性检验:表5参数估计值标准误差T统计量P 值结论C-7.2.-3.0.0018MA(2)的系数在10%的水平下显著，其余参数在5%的水平下显著.MA(1)0.0.3.0.0009MA(2)-0.0.-1.0.0557MA(3)-0.0.-4.0.0001MA(4)-0.0.-5.0.00002、拟合ARIMA(3,1,0)模型命令：LSD(X)CAR(1)AR(2)AR(3)在剔除不显著的C后得到估计结果(表6)表6

DependentVariable:D的Method:LeastSquaresDate05/05/10Time:22:49Sample(adjusted):1871193Bincludedobservations:68afteradjustments(1_B)Xt(1_B)XtCoefficientStdError t-StatisticProbAR(1)0.4951970.127277 389070900002AR(2)-0.2393320.135978 -1.7600740.0S31AR(3)-0,2555350.119253 -21420DO0.0369R-squared0.340051Meandependentvar-5.500000Adjusted^squared0.319745S.Ddependentvar82.26795S.E.ofregression67.85262Akaikeinfocritericn11.31567Sumsquaredresid299258.6Schwarzcriterion11.41359Loglikelihood-3817327卜hnn臼n-Quinncriter11.35447Durbin-Watsonstat1.866387InvertedARRoots45+641.46-641 -.41写出拟合的方程:(1-.4952B0.2393B2 0.2555B3)s.e.=67.85262 ,AIC=11.31567,SC=11.41359模型的显著性检验：CorrelogramofResiduals•q111¥is~iuri~rzraroincludedobservations;68Q-statisticprobabilitiesadjustedfor3ARMAterm(s)AutocorrelationPartialCorrelationACRACQ-StatProbII1II«1-0.011-0.0110.0084II1II112-0.042-0.0420.1354i2iIIJ130.0900.0890.7251I□11[14-0.08B-0.0B9130390.253I]11]150.0620.0701.59880451I[11c16-0.075-0.094203310.563II1II17-0.024-0.0012.07990.721I1II180.010-0J192.09320.937I[11119-0.05700322.35390.8841[11[110-0.094-0.1153.0S290.877'□1i匚111-0.142-0.1414.75660783'【1[112-0.051-0.06549795o.asa'I'1113-0.032-0.0455.06750.3871'1140.0210.025510470.92S1'II115C.0200.0045,14230.9531匚5i16-C.126-0.1376.60300.92111n170.2440.22812.139D.5951111118-C.001-0J3312.139图7根据图7,残差序列的Q统计量的p值均大于:=0.05，因此不拒绝原假设（原假设为:

序列为白噪声)，即残差序列是白噪声，拟合模型显著有效。模型参数的显著性检验：表7参数估计值标准误差T统计量P 值结论AR(1)0.0.3.0.0002所有参数在10%勺水平下显著AR(2)-0.0.-1.0.0831AR(3)-0.0.-2.0.03593、拟合ARIMA(3,1,4)模型命令：LSD(X)CAR(1)AR(2)AR(3)MA(1)MA(2)MA(3)MA(4)在剔除不显著的项后得到估计结果(表8)表8DependentVariable:D(X)Method：LeastSquaresDate;05J05/10Time;22:59Sample(adjusted)：10711938Includedobservations:68afteradjustmentsConvergenceachievedafter31iterationsMABackcast18571870CoefficientStd.Errort-StatisticProb.ARC2)-0.2209400,044143-5.1963440.0000AR(3)-0.6839400.059306-11.532450.0000MAC1)0.542640DJ295714.16953501.0001MAC2)0.46546901012104.6S86590.0000hi|A(3)0.9964700.03963611.116840.0000MA(4)0.3419290.1269212.69402100091R-sqjared0.410491Meandependentvar*5.500000AdjustedR-squared0.371595B,D.dependentvar62.26795S.E.ofregression65.21546Akaikeinfocriterion11.27737Sumsquaredresid2636396Schwarzcriterion11.47321Loglikelihood-377.4305Hannan-Quinncriter11.35496Durbin-Watsonstart2.010657InvertedARRoots.40+&4i40-841■79InvertedMARoots35-.93i.36+93i.41-83写出拟合的方程:(1_B)Xt(10.5428B0.4655B2 0.9965B2 0.3419B4)t(1_B)Xt(1+0.2289B2+0.6839B3)s.e.=65.21546 ，AIC=11.27737，SC=11.47321模型的显著性检验:

CorrelogramofResidualxDate:05/05/10Time:23:00Sample:16711938Includedobservations:58Q-statisticprobabilitiesadjustedfor6ARMAterm(s)AutocorrelationPartialCorrelationACPACQ-StatProb1[11111-0.032-003200713'□II1匚12-0.1144-0.1451.5709匸]■[!13-01166-0.1603.6000111|匚14<0.053-0.0963.80911111115007100104.193411116-0.039-Q.0924.31101t11[17-0.043-0.071445710.0351:1111801111009854262006611>1II90.02500085.477201401111