时间序列分析arma模型实验

1、基于ARMA1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ARMAI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2.1 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 。2.2 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

2、期发行或发生额扣除当期兑付或偿还额的差额。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ARMA真型构建3.1 判断序列的平稳性首先绘制出 M 的折线图，结果如下图：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

3、及异方差性，先对 m进行对数化处理，记为lm,其时序图如下:LM105-1QQ- 95- 9。-S.5- &0- 7S- 7J- 邑5-20-0S2007 20OS 20口920102011201220112014对数化后的趋势性减弱，但仍存在一定的趋势性，1、面观察lm的自相关图表3.1lm的自相关图Date: 11/02/14 T1me：2?:25Sample: 2005M11 2014M0SIncluded observations: 107AutocorrelationPartial CorrelationACPACQ-StatProb0 5290.52930,319110 0

4、000 574040967 45911121 HOUZJ1U 548101 091130 253U uuu1140.4470 015123.68ocoo11'60.410,071148.000.00D1二160.358-0.003162840.0001111JIJ1:178 g0 4220 3980.4010.1300.0950 101183 64202.09221.260.0000000000011:1100.43S0 109244.36000011110.373-o aia261 280 00011120 4970.192291.650.00011130 318'0.164

5、304 22000011140.330-0 090317.84000011150.267-0.142326.91o.coo11ie0.179-0 119331.0000001J1170.2540 071339 34ocoo111180 127西9341 440.00011190 1950.007346.02000n11200.1850.022350.6000011210.2300 144357.7600011,1:22230.2370 1770 029-0.027365.47309 840000.001J12403160.150383 39aoo1111:.Ji 口11111:1 111111

6、25262726293。0.1230.1110.094-0.0010 029-001B0.142-0 130-OOS9-0 059 HUB 口0 044336.04397 00399.09399 09389 22389.260.000000000000.000001 111111 " 1'匚11R "313233-0.0270.0010.0550.053-0.107-0 0130 1270.053339.37389.37389.0439Q.580000000000,00L00D上表口以乍1出，该lm序列的PACF只在滞舟-期、二期和三期是显著的,ACF随着滞后结束的

7、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。进一步进行单位根检验，由于存在较弱日勺趋势性且均值小为零，选择存衽趋势坝日勺形式，开根据AIC目动选择之后结束，单便根检验结果如下：表3.2单位根输出结果Null Hypothesis: LM has a unit rootExogenous: Constant, Linear TrendLag Length: 0 (Automatic - based on SIC, maxlag=12)Augmented Dickey-Fuller test statistic-8.6746460.0000Test critical values:1%

8、 level-4.0469255% level-3.45276410% level-3.151911t-StatisticProb.*MacKinnon (1996) one-sided p-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3.3 dlm曲线图观察dlm的自相关表：表3.3 dlm的自相关图Date: 11/02/14 Time: 22:35Sample: 2005M

9、11 2014M09Included observations: 106Autocorrelation Partial CorrelationAC PAC Q-Stat Prob*|.|*|.|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.|*|*|.|70.105-0.15643.56

10、30.000.|.|*|.|8-0.058-0.17143.9540.000.|.|*|.|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.

11、1170.2190.03784.2520.000*|.|.1.118-0.221-0.03690.6230.000.|*|.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.|.|.1.1260.0320.03

12、4124.230.000.|.|.1.1270.0590.037124.740.000*|.|.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*|.|*|.|35-0.215-0.197137.640.0

13、00.|*|.|*|360.3800.130161.260.000由dlm的自相关图可知，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3.4 sdlm的自相关图Date: 11/02/14 Time: 22:40Sample: 2005M11 2014M09Included observations: 94AutocorrelationPartial CorrelationACPACQ-StatProb*|.|*|.|1-0

14、.505-0.50524.7670.000.1.1*|.|2-0.057-0.41925.0820.000.1.1*|.|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.|*|.*|.|100.076-0.13939.90

15、20.000.|*|.|*|110.2270.24745.4850.000*|.|*|.|12-0.459-0.25968.6470.000.|*|*|.|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.

16、1.1.|.|20-0.052-0.03591.8410.000.|*|.|.|210.123-0.00993.7140.000.1.1.|*|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

17、.|.|290.0100.03297.9820.000.|*|.|.|300.1020.03999.4570.000.*|.|.*|.|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SAR

18、MA模型。3.2模型参数识别由表3.4 sdlm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，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)模型中选择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

19、) 12、SARMA(3,0) (1,1) 12 八个模型来拟合 sdlnm。3.3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差的检验，其他模型类似。回归结果为：表3.5 SARMA(1,0) (1,0) 12模型估计结果Dependent Variable: SDLMMethod: Least SquaresDate: 11/02/14 Time: 22:50Sample (adjusted): 2008M01 2014M09Included observations: 81 after adjustmentsConvergence achieved

20、after 6 iterationsVariableCoefficientStd. Errort-StatisticProb.C-0.0053050.023352-0.2271650.8209AR(1)-0.4908550.098580-4.9792560.0000SAR(12)-0.5485090.096987-5.6554710.0000R-squared0.448053Mean dependent var-0.004983Adjusted R-squared0.433901S.D. dependent var0.644876S.E. of regression0.485202Akaike

21、 info criterion1.427829Sum squared resid18.36280Schwarz criterion1.516512Log likelihood-54.82707Hannan-Quinn criter.1.463410F-statistic31.65901Durbin-Watson stat2.348799Prob(F-statistic)0.000000Inverted AR Roots.92+.25i.92-.25i.67+.67i.67-.67i.25-.92i.25+.92i-.25-.92i-.25+.92i-.49-.67-.67i-.67-.67i-

22、.92+.25i由表3.3可知，AR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3.6 SARMA(1,0) (1,0) 12模型的残差检验结果Date: 11/02/14 Time:Sample： 200SM01 2Q14M09Included cbservations: 010-statistic probabilities a<usledfor2 ARI.1 Aternn(s)Au

23、to co rrelationPartial CorrelationACPAC(ystetProbd d 1-0.181-0.1S12.7560Iil= 2*0374-0420U65Di i匚i30.075-0.12215,1400.000 i 40.124-0.05616 4820000|Q IE 5-0.137-0.15918 1450.0001p &0014-Q.0&418.15?a.001 n >70,1630.0&121,2020.001 i i a*00300.03021 2B30.0D2匚 i g-0145-0.03523 2460.002 i i

24、100 0500.00723,4340.003 j i ii0.047-0.02523700C.005匚匚i12-0172-QJ8026,5830.003 口 i 130 0850.00027 2880.CD4 nii 1 i140.1550.03628 7150.003ir i 匚 i15-0150-0.09232 0070.002ii ) '16OQOG0.082012a.0041 J111170 0B30.020327320.005 D "l '18-0.094-0.07333.6 阴0.006i ig*0 055-0.01 B33 9930.0D8i | ii

25、匚 i200.015-0.13034 0160.053i i | 210 0Q1-0.021M95 口0.014 J 12200790.13635,即0017 1 Ji230 0330.18135,7"0023匚 i匚i24-0 258-0.21643 5430.0D4i 1 i 1 2500310.01843 7570.0D6i | ii匚 i260017-0.162437920.0081 1 I E '2T0 022-012143 8510.0111匚1C 128-Q.1Q2-Q.21?45,1650.Q111 n ji2g0JS4QJ9949,521QQU51 1 1i

26、 300030-0.05649 6360.CD7匚 1i C 31-0 256-0.1U584460.00111i I 320 004-0.04658 4470.001 n i H hnn 、4 ucn nnn GFb"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SARMA模型的特征汇总表AICSC平稳性可逆性残差是否满 足白噪声SARMA(1,0) (1,0) 121.4278291.516512是是否SARMA(1,0) (1,1) 121.0954341.095434是是否SAR

27、MA(1,1) (1,0) 121.2061811.206181是是:是SARMA(1,1) (1,1) 120.8624961.010301是是P是SARMA(2,0) (1,0) 121.0103011.424354是是否SARMA(2,0) (1,1) 121.0002481.149124是是r否SARMA(3,0) (1,0) 121.2417641.391729是是r是SARMA(3,0) (1,1) 121.3917290.959325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3.8 SARMA(1,1) (

28、1,1) 12模型估计结果Dependent Variable: SDLMMethod: Least SquaresDate: 11/02/14 Time: 23:16Sample (adjusted): 2008M01 2014M09Included observations: 81 after adjustmentsConvergence achieved after 13 iterations MA Backcast: 2006M12 2007M12VariableCoefficientStd. Errort-StatisticProb.C-0.0068210.002943-2.3177

29、820.0232AR(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-squared0.701510Mean dependent var-0.004983Adjusted R-squared0.685800S.D. dependent var0.644876S.E. of regression0.361475Akaike info criterio

30、n0.862496Sum squared resid9.930500Schwarz criterion1.010301Log likelihood-29.93107Hannan-Quinn criter.0.921797F-statistic44.65381Durbin-Watson stat2.003373Prob(F-statistic)0.000000Inverted AR Roots.85+.23i.85-.23i.62-.62i.62+.62i.23+.85i.23-.85i.02-.23-.85i-.23+.85i-.62+.62i-.62+.62i-.85-.23i-.85+.2

31、3iInverted MA Roots.99.86+.49i.86-.49i.83.49-.86i.49+.86i.00-.99i-.00+.99i-.49-.86i-.49+.86i-.86-.49i-.86+.49i-.9920M3.2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月的序列值，并将结果保存在sdlnmf中，预测情况如下：For&cast SDLMF Actual: SDLMForecasl sample: S014M05 S014M09Included observations: 5Root Mean Square

32、d Error 0.648539Mean Absolute Error 0.461327Mean Abs. Percent Error 62.91846Theil Inequality Goefficient 0.53&1 MBias Proporttori0.000107Variance Proportion 0.64 9319 Covariance Proportton 0350574SDLMF TiSE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bias proportion表示偏误，即预测均值与真实均值的偏离程度，本例中 bias proportion的值

33、为0.000107,预 测均值与真实值偏离较小；variance proportion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variance proportion为0.649319,则说明预测波动与真实波动的差异较大； covariance proportion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指标社会融资规模地区全国频度月单位亿元2002-01-4722002-022892002-0331362002-0411512002-0

34、517742002-0626212002-078132002-0815852002-0935072002-107952002-1118052002-1231092003-0133862003-029982003-0340412003-0426222003-0529712003-0658422003-0713442003-0833212003-0940402003-1012182003-1118322003-1224982004-0121142004-024382004-0365572004-0427312004-0524432004-0632292004-075902004-081501200

35、4-0929812004-104832004-1119772004-1235862005-0136202005-028242005-0341892005-0419992005-0519682005-0647232005-076292005-0820972005-0960412005-10-9742005-1123682005-1225242006-0163232006-0217372006-0374722006-0433252006-0537852006-0638432006-0722542006-0833622006-0930772006-108942006-1127882006-12383

36、72007-0169082007-0230832007-0363112007-0461032007-0538242007-0670422007-0731002007-0869612007-0952902007-1036882007-1130732007-1242812008-01108592008-0247312008-0363912008-0470762008-0556782008-0659762008-0748902008-0845752008-0956592008-1012882008-1145172008-1281642009-01139902009-02111312009-03220

37、112009-0454522009-05149592009-06210672009-0773882009-0876502009-09118712009-1059852009-1195012009-1281002010-01205502010-02108772010-03138302010-04149192010-05108052010-06101962010-0772022010-08106462010-09112242010-1086082010-11105542010-12107802011-01175602011-0264682011-03182122011-04136732011-05

38、108542011-06108732011-0753932011-08107412011-0942792011-1079082011-1195812011-12127442012-0197542012-02104312012-03187042012-0496372012-05114322012-06178022012-07105222012-08124752012-09164622012-10129062012-11112252012-12162822013-01254462013-02107052013-03255032013-04176292013-05118712013-06103752013-0781912013-08158412013-09141202013-1086452013-11123102013-12125322014-0126003.942014-029369.772014-0320934.492014-0415259.452014-0514013.272014-0619673.172014-072736.942014-089576.522014-0910522.06存在问题本次应用ARMA模型分析数据的过程存在不少问题，在整个过程中感觉对模型的理解 还不够深入，有一些细节没有理解清楚，