时间序列分析——ARMA模型实验讲解

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：- # -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

3、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lm,其时序图如 下:LM2005 20DG 20072008200920102011201220132014图3.2 lm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3.1 1m的自相关图Date: 11/02H4 Time: 22:25 Sample: 2005M11 2014M09 Included observations: 1071 111111111口10.5290.62930.8190,00020.5740.40967,4590,00030.5480.2531。1,口90.000&#

4、39;114 0 447 0.016 123.68 0,0001 11115 0.461 0 071 148 00 0.0001 二1 16 0.358 -0 063 162.84 0.000 Tn1117 0.422 0.130 133.64 0.000 二118 0,396 0.095 202.08 0.000II11ZH9 0,401 0 101 221,26 0,0001 111Ji10 0.438 0.109 244.36 0,0001 11111 0,373 -0.018 261.28 0 0001_J112 0 497 0 192 291 65 0 000i _1!113 0.3

5、18 -0 164 304.22 0.0001 口1 114 0.330 -0.090 317.84 0.000'ZJ1匚115 0.267 -0.142 326.91 0.0001 0i |Ei16 0,179 -0,119 331 00 0.0001 ZJi117 0.254 0.071 339,34 0,0001 口111118 0,127 -0.059 341.44 0,0001119 0 185 0.007 346.02。口L1 1120 0.1S5 0.022 350 60 0.00'11121 0.230 0 144 357.76 0.001 =11 122 0.

6、237 0 028 36547 0.00> 口1123 0 177 -0.027 369.84 0.001 o1l24 0316 0,150 383 09 0.001 JI125 0 123 -0 142 366.04 0.001 JIIC12B 0.111 -0.130 387.80 0.001 : 11 127 0,094 -0.058 389.09 0.00i 11 1128 -0.001 -0 059 389 09 0.00> 1 )1 c129 0.029 -O OBO 389.22 0.001 1 11130 -0.016 0 044 389.26 0.001 1 1i

7、r131 -0.027 -0.107 389.37 0.000'111132 0,001 -0,013 39937 0,000> U 11JI33 0,055 0.127 369,84 0,0001 11134 0.0&8 0.053 390.58 0,000Autocorrelation Partial CorrelationAC PAC Q-Stat ProbACFM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

8、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Null Hypothesis: LM has a unit rootExogenous: Constant, Linear TrendLag Length: 0 (Automatic - based on SIC, maxlag=12)t-StatisticProb.*Augmented Dickey-Fuller test statistic-8.6746460.0000Test critical values:1% level-4.0469255% level-3.45276410% level-3.151

9、911*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: 2005M11 2014M09Included observations: 106Autocorrelation Partial C

10、orrelationAC 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.|*|*|.|70.105-0.15643.5630.000.|.|*|.|8-0.058-0.17143.9540.000.|.|*|.|9-0.019-0.196

11、43.9960.000.|*|.|.|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*|.|.1.118-0.221-0.03690.6230.000.|

12、*|.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*|.|.1.128-0.126

13、0.044127.080.000.|*|*|.|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，dlm在滞后期为12、24、

14、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*|.|*|.|1-0.505-0.50524.7670.000.1.1*|.|2-0.057-0.41925.0820.000.1.1

15、*|.|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*|.|*|.|12-0.459-0.2

16、5968.6470.000.|*|*|.|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.|*|.|.|210.123-0.00993.7

17、140.000.1.1.|*|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.|*|.|.|300.1020.03999.4570.00

18、0.*|.|.*|.|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。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

19、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。3.3 模型参数估计以SARMA(

20、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 after 6 iterationsVariableCoefficientStd. Errort-Statis

21、ticProb.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 info criterion1.427829Sum squared resid18.36280Schwarz

22、 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-.92+.25i-.92-.25i由表3.3可知，AR(1)与sar ()的P值均小于0.05,参数显著，可以

23、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: 22:54Sample: 2008M01 2014MD9Included observations： 81-statistic probabilities adjusted for 2 ARMA termAutocorrelation Partial Correlation AC PAC Q-Stat

24、 ProbI匚III匚IIIIII|11 11 l|匚二III11匚I111I11111>11 -0.181-0.181-0.420-0.122-0.056-0.159-0.0640.0910.030275601465015.1401648216,14510,16221,20221,2830.0000.0000.0000 0010.0010.0022345678-037400750.124-0137-0.0140.183-0030I匚11119-0 145-0.032232460.002I1«1000500.007234840.003I11>110 047-0.02523

25、.7000.005I匚1ci12-0172-0.180265330.003I1IIi130.0850.00027,2680.004IJ'1| i140.1550.03629,7130.003I匚1i Li15-0.150-0.09232,0070 002I111160 0060.082320120.004I1111700330.020327320.005 r11 113-0094-0.073336660.006 i11 1119-0055-0.016339930.00BII1IC1200015-0.130340160.013I111210.091-0.03134,9500.014I1

26、"II 1220.0790.13635,6540.017I1 1112300330.18135,7770 023匚1124-0258-0.216436430.004I1112500310.01S437570.006I11匚1260.017-0.162437920.00BII11匚1270022-0.12143 8510.011(E1匚128*0,102-0.21345,1660011I112 290.1840.09949,5210.005I11 1300.030-0.05649,6360 0071)E131-0256-0.1145B4460.001IIi1 ki 1 l L1II32

27、r r0 004ft 4 Hi e-0.046A 4 AC58 447en ftn a0.001n. 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_ _,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P是_ _,、12SARMA(1

28、,1) (1,1)0.8624961.010301是是是_ _ _,_、12SARMA(2,0) (1,0)1.0103011.424354是是否一，、12SARMA(2,0) (1,1)1.0002481.149124是是否_ _ _,_、12SARMA(3,0) (1,0)1.2417641.391729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3.8 SARMA(1,1) (1,1) 12模型估计结果Dependent Var

29、iable: 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-0.0068210.002943-2.3177820.0232AR(1)0.0186630.141

30、1680.1322030.8952SAR(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 criterion0.862496Sum squared resid

31、9.930500Schwarz 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+.23iInverted MA Roots.99.86+

32、.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 ?2 S.E.值，并将结果保存在 sdlnmf中，预测情况如下：Forecast: SDLMFActual： SDLL1Forecast sample: 2014M05 2014M09Included observations: 5Root Mean Squared Error0.648539Me

33、an Absolute Error0.461327Mean Abs. Percent Error62.91846Theil Inequality Coefficient0 538154Bias Proportion0.000107Variance Proportion0.649319Covariance Proportion0.350574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的值为0.000107,预 测均值与真实值偏离较小；variance propor

34、tion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2002-078132003-052971地区全国2002-0815852003-065842频度月2002-0935072003-071344单位亿元2002-107952003-0833212002

35、-01-4722002-1118052003-0940402002-022892002-1231092003-1012182002-0331362003-0133862003-1118322002-0411512003-029982003-1224982002-0517742003-0340412004-0121142002-0626212003-0426222004-02438-11 -2004-0365572004-0427312004-0524432004-0632292004-075902004-0815012004-0929812004-104832004-1119772004-12

36、35862005-0136202005-028242005-0341892005-0419992005-0519682005-0647232005-076292005-0820972005-0960412005-10-9742005-1123682005-1225242006-0163232006-0217372006-0374722006-0433252006-0537852006-0638432006-0722542006-0833622006-0930772006-108942006-1127882006-1238372007-0169082007-0230832007-03631120

37、07-0461032007-0538242007-0670422007-0731002007-086961|2007-0952902007-1036882007-1130732007-1242812008-01108592008-0247312008-0363912008-0470762008-0556782008-0659762008-0748902008-0845752008-0956592008-1012882008-1145172008-1281642009-01139902009-02111312009-03220112009-0454522009-05149592009-06210

38、672009-0773882009-0876502009-09118712009-1059852009-1195012009-1281002010-01205502010-02108772010-03138302010-04149192010-05108052010-06101962010-0772022010-08106462010-09112242010-1086082010-11105542010-12107802011-01175602011-0264682011-03182122011-04136732011-05108542011-06108732011-0753932011-08

39、107412011-0942792011-1079082011-1195812011-12127442012-0197542012-02104312012-03187042012-0496372012-05114322012-06178022012-07105222012-08124752012-09164622012-10129062012-11112252012-12162822013-01254462013-02107052013-03255032013-04176292013-05118712013-06103752013-0781912013-08158412013-09141202013-1086452013-11123102013-12125322014-0126003.942014-029369.772014-0320934.492014-0415259.452014-0514013.272014-0619673.172014-072736.942014-089576.52|2014-0910522.06存在问题本次应用ARMA模型分析数据的过程存