时间序列分析——ARMA模型试验

1、基于ARMA模型的社会融资规模增长分析ARM模型实验第一部分实验分析目的及方法一般说来,若时间序列满足平稳随机过程的性质,则可用经典的ARMAI型进行建模和预则。但是,由于金融时间序列随机波动较大，很少满足ARMAg型的适用条件，无法直接采用该模型进行处理。通过对数化及差分处理后，将原本非平稳的序列处理为近似平稳的序列,可以采用ARM喂型进行建模和分析。第二部分实验数据数据来源数据来源于中经网统计数据库。具体数据见附录表5.1。所选数据变量社会融资规模指一定时期内（每月、每季或每年）实体经济从金融体系获得的全部资金总额，为一增量概念，即期末余额减去期初余额的差额，或当期发行或发生额扣除当期兑付

2、或偿还额的差额。社会融资规模作为重要的宏观监测指标，由实体经济需求所决定，反映金融体系对实体经济的资金量支持。本实验拟选取2005年11月到2014年9月我国以月为单位的社会融资规模的数据来构建ARM喂型，并利用该模型进行分析预测。第三部分ARMA1型构建判断序列的平稳性首先绘制出M的折线图，结果如下图:30,000-1图3.1社会融资规模M曲线图从图中可以看出，社会融资规模M序列具有一定的趋势性，由此可以初步判断该序列是非平稳的。此外，m在每年同时期出现相同白变动趋势，表明m还存在季节特征。下面对m的平稳性和季节性进行进一步检验。为了减少m的变动趋势以及异方差性，先对m进行对数化处理，记为l

3、m,其时序图如下:LM200520DG20072008200920102011201220132014图3.2lm曲线图1m的自相关图对数化后的趋势性减弱，但仍存在一定的趋势性，下面观察表3.11m的自相关图Date:11/02H4Time:22:25Sample:2005M112014M09Includedobservations:1071111111111口10.5290.62930.8190,00020.5740.40967,4590,00030.5480.2531。1,口90.000'11404470.016123.680,0001111150.4610071148000.00

4、01二1160.358-0063162.840.000Tn11170.4220.130133.640.000二1180,3960.095202.080.000II11ZH90,4010101221,260,0001111Ji100.4380.109244.360,0001111110,373-0.018261.2800001_J11204970192291650000i_1!1130.318-0164304.220.0001口11140.330-0.090317.840.000'ZJ1匚1150.267-0.142326.910.00010i|Ei160,179-0,119331000

5、.0001ZJi1170.2540.071339,340,0001口1111180,127-0.059341.440,000111901850.007346.02。口L111200.1S50.022350600.00'111210.2300144357.760.001=111220.2370028365470.00>口11230177-0.027369.840.001o1l2403160,150383090.001JI1250123-0142366.040.001JIIC12B0.111-0.130387.800.001:111270,094-0.058389.090.00i11

6、1128-0.001-0059389090.00>1)1c1290.029-OOBO389.220.001111130-0.0160044389.260.00111ir131-0.027-0.107389.370.000'1111320,001-0,013399370,000>U11JI330,0550.127369,840,0001111340.0&80.053390.580,000AutocorrelationPartialCorrelationACPACQ-StatProbACF!着滞后结上表可以看出，该lm序列的PAC次在滞后一期、二期和三期是显著的,进一步

7、进行单位根检验，AIC自动选择之后结束的增加慢慢衰减至0,由此可以看出该序列表现出一定的平稳性。由于存在较弱的趋势性且均值不为零，选择存在趋势项的形式，并根据束，单位根检验结果如下:表3.2单位根输出结果NullHypothesis:LMhasaunitrootExogenous:Constant,LinearTrendLagLength:0(Automatic-basedonSIC,maxlag=12)t-StatisticProb.*AugmentedDickey-Fullerteststatistic-8.6746460.0000Testcriticalvalues:1%level-4.

8、0469255%level-3.45276410%level-3.151911*MacKinnon(1996)one-sidedp-values.单位根统计量ADF=-8.674646小于临界值，且P为0.0000,因此该序列不存在单位根,即该序列是平稳序列。由于趋势性会掩盖季节性，从lm图中可以看出，该序列有一定的季节性，为了分析季节性，对lm进行差分处理，进一步观察季节性：DLM图3.3dlm曲线图观察dlm的自相关表：表3.3dlm的自相关图Date:11/02/14Time:22:35Sample:2005M112014M09Includedobservations:106Autoco

9、rrelationPartialCorrelationACPACQ-StatProb*|.|*|.|1-0.566-0.56634.9340.000.|*|*|.|20.113-0.30536.3410.000.|.|*|.|30.032-0.09336.4550.000*|.|*|.|4-0.084-0.11437.2440.000.|*|.|.|50.1050.01538.4940.000*|.|*|.|6-0.182-0.18242.2960.000.|*|*|.|70.105-0.15643.5630.000.|.|*|.|8-0.058-0.17143.9540.000.|.|*|.

10、|9-0.019-0.19643.9960.000.|*|.|.|100.110-0.04545.4290.000*|.|*|.|11-0.242-0.32952.5010.000.|*|.1.1120.3630.02368.5160.000*|.|.1.113-0.2020.03273.5340.000.|*|.|*|140.1010.12574.8150.000.|.|.|*|150.0040.14174.8170.000*|.|*|.|16-0.161-0.08978.1100.000.|*|.1.1170.2190.03784.2520.000*|.|.1.118-0.221-0.03

11、690.6230.000.|*|.1.1190.089-0.04691.6620.000*|.|*|.|20-0.080-0.15892.5160.000.|.|.1.1210.067-0.03993.1150.000.|.|.1.1220.0680.05693.7490.000*|.|*|.|23-0.231-0.130101.080.000.|*|.|*|240.3590.116119.040.000*|.|.|*|25-0.1890.123124.090.000.|.|.1.1260.0320.034124.230.000.|.|.1.1270.0590.037124.740.000*|

12、.|.1.128-0.1260.044127.080.000.|*|*|.|290.087-0.079128.210.000.|.|.|*|30-0.0500.092128.580.000.|.|.1.131-0.037-0.019128.790.000.|.|*|.|32-0.035-0.113128.970.000.|.|.1.1330.041-0.056129.240.000.|*|.1.1340.078-0.027130.210.000*|.|*|.|35-0.215-0.197137.640.000.|*|.|*|360.3800.130161.260.000由dlm的自相关图可知，

13、dlm在滞后期为12、24、36等差的自相关系数均显著异于零。因此该序列为以12为周期呈现季节性，而且季节自相关系数并没有衰减至零，因此为了考虑这种季节性，进行季节性差分，得新变量sdlm:观察sdlm的自相关图：表3.4sdlm的自相关图Date:11/02/14Time:22:40Sample:2005M112014M09Includedobservations:94AutocorrelationPartialCorrelationACPACQ-StatProb*|.|*|.|1-0.505-0.50524.7670.000.1.1*|.|2-0.057-0.41925.0820.000.

14、1.1*|.|30.073-0.29225.6090.000.|*|.|.|40.1600.06728.1690.000*|.|.*|.|5-0.264-0.12535.2520.000.|*|.*|.|60.098-0.11036.2440.000.|*|.|.|70.0980.01937.2430.000.|.|.|*|8-0.0410.08237.4190.000.*|.|.|.|9-0.132-0.03839.2750.000.|*|.*|.|100.076-0.13939.9020.000.|*|.|*|110.2270.24745.4850.000*|.|*|.|12-0.459-

15、0.25968.6470.000.|*|*|.|130.193-0.25172.7770.000.|*|.*|.|140.132-0.10174.7530.000.*|.|.*|.|15-0.142-0.18977.0560.000.1.1.|.|16-0.053-0.05677.3780.000.|*|.|*|170.2330.09183.7510.000*|.|.*|.|18-0.234-0.17990.2580.000.|*|.|.|190.1020.05491.5050.000.1.1.|.|20-0.052-0.03591.8410.000.|*|.|.|210.123-0.0099

16、3.7140.000.1.1.|*|22-0.0590.12094.1500.000.1.1.|*|23-0.0110.21594.1660.000.1.1.*|.|24-0.032-0.17094.3010.000.|*|.*|.|250.088-0.13795.3030.000.*|.|.|.|26-0.105-0.03496.7600.000.|*|.*|.|270.077-0.11697.5620.000.1.1.*|.|28-0.054-0.17897.9670.000.1.1.|.|290.0100.03297.9820.000.|*|.|.|300.1020.03999.4570

17、.000.*|.|.*|.|31-0.179-0.099104.060.000.1.1.|.|320.071-0.058104.790.000.1.1.*|.|330.031-0.066104.930.000.*|.|.*|.|34-0.089-0.144106.130.000.|.|.|*|350.0360.082106.320.000.|*|.*|.|360.105-0.102108.050.000Sdlm在滞后期24之后的季节ACF和PACF已衰减至零，下面对sdlm建立SARMA模型。模型参数识别由表3.4sdlm的自相关图的自相关图可知，偏自相关系数在3阶后都落在两倍标准差的范围以内

18、，即不显著异于零。自相关系数在1阶和12阶显著异于零。因此SARMA(p,q莫型中选择p、q均不超过3。此外，由于高阶移动平均模型估计较为困难而且自回归模型可以表示无穷阶的移动平均过程，因此Q尽可能取小。拟选择SARMA(1,0)(1,0)12、SARMA(1,0)(1,1)12、SARMA(1,1)(1,0)12、SARMA(1,1)(1,1)12、SARMA(2,0)(1,0)12、SARMA(2,0)(1,1)12、SARMA(3,0)(1,0)12、SARMA(3,0)(1,1)12八个模型来拟合sdlnm。模型参数估计以SARMA(1,0)(1,0)12模型为例，分析该模型的估计及残

19、差的检验，其他模型类似。回归结果为：表3.5SARMA(1,0)(1,0)12模型估计结果DependentVariable:SDLMMethod:LeastSquaresDate:11/02/14Time:22:50Sample(adjusted):2008M012014M09Includedobservations:81afteradjustmentsConvergenceachievedafter6iterationsVariableCoefficientStd.Errort-StatisticProb.C-0.0053050.023352-0.2271650.8209AR(1)-0.4

20、908550.098580-4.9792560.0000SAR(12)-0.5485090.096987-5.6554710.0000R-squared0.448053Meandependentvar-0.004983AdjustedR-squared0.433901S.D.dependentvar0.644876S.E.ofregression0.485202Akaikeinfocriterion1.427829Sumsquaredresid18.36280Schwarzcriterion1.516512Loglikelihood-54.82707Hannan-Quinncriter.1.4

21、63410F-statistic31.65901Durbin-Watsonstat2.348799Prob(F-statistic)0.000000InvertedARRoots.92+.25i.92-.25i.67+.67i.67-.67i.25-.92i.25+.92i-.25-.92i-.25+.92i-.49-.67-.67i-.67-.67i-.92+.25i-.92-.25i由表3.3可知，AR(1)与sar(12)的P值均小于0.05,参数显著，可以通过检验。该模型AIC为1.427829,SC值为1.516512。回归结果的最后一部分表示该模型滞后多项式的反特征根，小于1,因此

22、该模型是平稳的。卜面对残差进行检验。观察残差的自相关图:表3.6SARMA(1,0)(1,0)12模型的残差检验结果Date.11/02/14Time:22:54Sample:2008M012014MD9Includedobservations：81-statisticprobabilitiesadjustedfor2ARMAtermAutocorrelationPartialCorrelationACPACQ-StatProbI匚III匚IIIIII|1111l|匚二III11匚I111I11111>11-0.181-0.181-0.420-0.122-0.056-0.159-0.06

23、40.0910.030275601465015.1401648216,14510,16221,20221,2830.0000.0000.00000010.0010.0022345678-037400750.124-0137-0.0140.183-0030I匚11119-0145-0.032232460.002I1«1000500.007234840.003I11>110047-0.02523.7000.005I匚1ci12-0172-0.180265330.003I1IIi130.0850.00027,2680.004IJ'1|i140.1550.03629,7130.

24、003I匚1iLi15-0.150-0.09232,0070002I1111600060.082320120.004I1111700330.020327320.005r11113-0094-0.073336660.006i111119-0055-0.016339930.00BII1IC1200015-0.130340160.013I111210.091-0.03134,9500.014I1"II1220.0790.13635,6540.017I11112300330.18135,7770023匚1124-0258-0.216436430.004I1112500310.01S43757

25、0.006I11匚1260.017-0.162437920.00BII11匚1270022-0.121438510.011(E1匚128*0,102-0.21345,1660011I112290.1840.09949,5210.005I111300.030-0.05649,63600071)E131-0256-0.1145B4460.001IIi1ki1lL1II32rr0004ft4Hie-0.046A4AC58447enftna0.001n.ncT由表3.6可知，由Q统计量可知残差存在自相关性，P值远小于0.05,因此残差不满足白噪声的假设。将八个模型的估计结果进行汇总如下:表3.7不同S

26、ARMA模型的特征汇总表AICSC平稳性可逆性残差是否满足白噪声_,12SARMA(1,0)(1,0)1.4278291.516512是是否_，、12SARMA(1,0)(1,1)1.0954341.095434是是否_，一、12SARMA(1,1)(1,0)1.2061811.206181是是是_,、12SARMA(1,1)(1,1)0.8624961.010301是是是_,_、12SARMA(2,0)(1,0)1.0103011.424354是是否_、12SARMA(2,0)(1,1)1.0002481.149124是是否_,_、12SARMA(3,0)(1,0)1.2417641.391

27、729是：是是_,、12SARMA(3,0)(1,1)1.3917290.959325是是是综合来看，根据信息准则，应选择SARMA(1,1)(1,1)12对数据进行拟合是最优的。拟合结果为：表3.8SARMA(1,1)(1,1)12模型估计结果DependentVariable:SDLMMethod:LeastSquaresDate:11/02/14Time:23:16Sample(adjusted):2008M012014M09Includedobservations:81afteradjustmentsConvergenceachievedafter13iterationsMABackc

28、ast:2006M122007M12VariableCoefficientStd.Errort-StatisticProb.C-0.0068210.002943-2.3177820.0232AR(1)0.0186630.1411680.1322030.8952SAR(12)-0.2016230.120638-1.6713130.0988MA(1)-0.8339470.080352-10.378650.0000SMA(12)-0.8603910.041002-20.984270.0000R-squared0.701510Meandependentvar-0.004983AdjustedR-squ

29、ared0.685800S.D.dependentvar0.644876S.E.ofregression0.361475Akaikeinfocriterion0.862496Sumsquaredresid9.930500Schwarzcriterion1.010301Loglikelihood-29.93107Hannan-Quinncriter.0.921797F-statistic44.65381Durbin-Watsonstat2.003373Prob(F-statistic)0.000000InvertedARRoots.85+.23i.85-.23i.62-.62i.62+.62i.

30、23+.85i.23-.85i.02-.23-.85i-.23+.85i-.62+.62i-.62+.62i-.85-.23i-.85+.23iInvertedMARoots.99.86+.49i.86-.49i.83.49-.86i.49+.86i.00-.99i-.00+.99i-.49-.86i-.49+.86i-.86-.49i-.86+.49i-.993.2模型预测在SARMA(1,1)(1,1)12估计方程下选择动态估计，预测2014年10月至12月的序列SDLMF-?2S.E.值，并将结果保存在sdlnmf中，预测情况如下：Forecast:SDLMFActual：SDLL1Fo

31、recastsample:2014M052014M09Includedobservations:5RootMeanSquaredError0.648539MeanAbsoluteError0.461327MeanAbs.PercentError62.91846TheilInequalityCoefficient0538154BiasProportion0.000107VarianceProportion0.649319CovarianceProportion0.350574图中左边是预测值与置信区间，右边是预测的误差。Theil不等系数中biasproportion表示偏误，即预测均值与真实均

32、值的偏离程度，本例中biasproportion的值为0.000107,预测均值与真实值偏离较小；varianceproportion表示方差误，用来反映预测波动与真实波动之间的差异，本例varianceproportion为0.649319,则说明预测波动与真实波动的差异较大；covarianceproportion表示协方差误，反映残存非系统性预测误差，本例中该值为0.350574,该误差占比越大，预测效果越好。本例中的协方差误要小于方差误，因此预测效果较差。附录具体数据表5.1社会融资规模M指标社会融资规模2002-078132003-052971地区全国2002-0815852003-

33、065842频度月2002-0935072003-071344单位亿元2002-107952003-0833212002-01-4722002-1118052003-0940402002-022892002-1231092003-1012182002-0331362003-0133862003-1118322002-0411512003-029982003-1224982002-0517742003-0340412004-0121142002-0626212003-0426222004-024382004-0365572004-0427312004-0524432004-0632292004-

34、075902004-0815012004-0929812004-104832004-1119772004-1235862005-0136202005-028242005-0341892005-0419992005-0519682005-0647232005-076292005-0820972005-0960412005-10-9742005-1123682005-1225242006-0163232006-0217372006-0374722006-0433252006-0537852006-0638432006-0722542006-0833622006-0930772006-1089420

35、06-1127882006-1238372007-0169082007-0230832007-0363112007-0461032007-0538242007-0670422007-0731002007-086961|2007-0952902007-1036882007-1130732007-1242812008-01108592008-0247312008-0363912008-0470762008-0556782008-0659762008-0748902008-0845752008-0956592008-1012882008-1145172008-1281642009-011399020

36、09-02111312009-03220112009-0454522009-05149592009-06210672009-0773882009-0876502009-09118712009-1059852009-1195012009-1281002010-01205502010-02108772010-03138302010-04149192010-05108052010-06101962010-0772022010-08106462010-09112242010-1086082010-11105542010-12107802011-01175602011-0264682011-031821

37、22011-04136732011-05108542011-06108732011-0753932011-08107412011-0942792011-1079082011-1195812011-12127442012-0197542012-02104312012-03187042012-0496372012-05114322012-06178022012-07105222012-08124752012-09164622012-10129062012-11112252012-12162822013-01254462013-02107052013-03255032013-04176292013-05118712013-06103752013-0781912013-08158412013-09141202013-1086452013-11123102013-12125322014-0126003.942014-029369.772014-0320934.492014-0415259.452014-0514013.272014-0619673.172