时间序列分析ARMA模型实验

1、基于ARM模型的社会融资规模增长分析ARMA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ARM/模型进行建模 和预则。但是,由于金融时间序列随机波动较大,很少满足ARM/模型的适用条件,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ARMA模型进行建模和分析。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，或当期发行

2、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ARMA模型，并利用该模型进行分析预测。第三部分ARMA模型构建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的变动趋势，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

3、行对数化处理，记为Im,其时序图如下:图3.2 Im曲线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 lm的自相关图lm的自相关图Date: 11/02/14 Time 22:25Sample; 2005H11 2014M03Included observations: 107Auto correlation Partial CorrelationAC PAC Q-Slat"InnnndjnlnnrlnI110 5290.52930.S190 000I120 5740409674590000I二3'0.5490 253101 090000I1斗0.4470

4、 016123.630000I'5'0|.4£103114S.00o.coo'D6'0.35S-0僦316ZS40X00I3110 4220.130133 640.000IJ 1S'01.3960.095202 OS0000Ig'0.4010 101221.250000Ili100.4300 109244.36ocooI1110.S73-0 01326123ocooI1Z0 4970.192291 650000I口1130 318'0 ie4304 220000>114'0.330'-0 090317 04

5、0000'115'0.267-0.142325.910.000I口10.179-0 119331.000000I1170.2540 a7i339.34ocooI B11G0 '1270.059341 440.000I119'0 1360.007346.02QUO"I120'0.1350 022s&o.eo000IF210.2300 144357.76000I1 '220.2370 023365 47000I 1230 177-0.027309 040001J'240 3160 15033389aoo1 口125.123-

6、0J42336.040.00IE1260.111-0 130397.000001 B1270.094-oasQ399 090001 (12&-O.001-0 a59309 090001 G129'0 029-0.030339 220.0011300160 04433920aoo1口131-0.027-0 1073391.37000011320.001-0 013339.37ocoo1：33'0.0550 127339 54ocoo134'0.06S0,053390,530X00二期和三期是显著的, 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

7、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：上表可以看出，该Im序列的PACf只在滞后一期、ACF随着滞后结进一步进行单位根检验，AIC自动选择之后结表3.2单位根输出结果Null Hypo thesis: LM has a unit rootExogenous: Constant, Linear TrendLag Length: 0 (Automatic - based on SIC, maxlag=12)t-StatisticProb.*Augmented Dickey-Fuller test statistic-8.6746460.0000Tes

8、t critical values:1% level-4.0469255% level-3.45276410% level-3.151911*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，为了分观察dlm的自相关表:表3.3 dlm的自相关图Date: 11/02/14Time: 22:35Sam pie: 2005M11 2014M09Included observations: 10

9、6AutocorrelationP artial CorrelationACPACQ-StatProb*|.I*|.I1-0.566-0.56634.9340.000.|*l*|.I20.113-0.30536.3410.000.|.I*|.I30.032-0.09336.4550.000*|.I*|.I4-0.084-0.11437.2440.000|* |LI50.1050.01538.4940.000*|-|1I6-0.182-0.18242.2960.000|* |1I70.105-0.15643.5630.000|. |1I8-0.058-0.17143.9540.000|. |1I

10、9-0.019-0.19643.9960.000|* |LI100.110-0.04545.4290.000*|. |*|.I11-0.242-0.32952.5010.000|* |LI120.3630.02368.5160.0001 |LI13-0.2020.03273.5340.000|* |I*I140.1010.12574.8150.000LlI*I150.0040.14174.8170.0001l1I16-0.161-0.08978.1100.000l*lLI170.2190.03784.2520.000*|. |LI18-0.221-0.03690.6230.000l*lLI19

11、0.089-0.04691.6620.0001l1I20-0.080-0.15892.5160.000LlLI210.067-0.03993.1150.000LlLI220.0680.05693.7490.000*|. |1I23-0.231-0.130101.080.000l* lI*I240.3590.116119.040.0001lI*I25-0.1890.123124.090.000LlLI260.0320.034124.230.000LlLI270.0590.037124.740.0001lLI28-0.1260.044127.080.000l*l1I290.087-0.079128

12、.210.000LlI*I30-0.0500.092128.580.000LlLI31-0.037-0.019128.790.000LlII32-0.035-0.113128.970.000LlLI330.041-0.056129.240.000l*lLI340.078-0.027130.210.000*|. |II35-0.215-0.197137.640.000I*|I*I360.3800.130161.260.000由dim的自相关图可知，dim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

13、节性，进行季节性差分，得新变量sdim :观察sdim的自相关图:表3.4 sdim的自相关图Date: 11/02/14 Time: 22:40Sam pie: 2005M11 2014M09Inciuded observations: 94AutocorrelationP artial CorrelationACPACQ-StatProb*| |*| |1-0.505-0.50524.7670.000.|. |*|. |2-0.057-0.41925.0820.000.|. |*|. |30.073-0.29225.6090.000.|* |.|. |40.1600.06728.1690.

14、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*|. |*|. |12-0.459-0.25968.6470.000.|* |*|. |130.193-0.25172

15、.7770.000.|* |.*|. |140.132-0.10174.7530.000.*|. |.*|. |15-0.142-0.18977.0560.000.|. |.|. |16-0.053-0.05677.3780.000.|* |.|* |170.2330.09183.7510.000*|. |.*|. |18-0.234-0.17990.2580.000.|* |.|. |190.1020.05491.5050.000.|. |.|. |20-0.052-0.03591.8410.000.|* |.|. |210.123-0.00993.7140.000.|. |.|* |22-

16、0.0590.12094.1500.000.|. |.|* |23-0.0110.21594.1660.000.|. |.*|. |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.|. |.*|. |28-0.054-0.17897.9670.000.|. |.|. |290.0100.03297.9820.000.|* |.|. |300.1020.03999.4570.000.

17、*|. |.*|. |31-0.179-0.099104.060.000.|. |.|. |320.071-0.058104.790.000.|. |.*|. |330.031-0.066104.930.000.*|. |.*|. |34-0.089-0.144106.130.000.|. |.|* |350.0360.082106.320.000.|* |.*|. |360.105-0.102108.050.000Sdim在滞后期24之后的季节 ACF和PACF已衰减至零，下面对 sdim建立SARMA模型。3.2模型参数识别由表3.4 sdim的自相关图的自相关图可知，偏自相关系数在3阶后

18、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)仁、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(1,0

19、)( 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Dep endent Variable: SDLMMethod: Least SquaresDate: 11/02/14 Time: 22:50Samp le (adjusted): 2008M01 2014M09Included observations: 81 after adjustmentsConvergence achieved after 6 iterationsVariableCoefficientStd. Errort-Statistic

20、P rob.C-0.0053050.023352-0.2271650.8209AR(1)-0.4908550.098580-4.9792560.0000SAR(12)-0.5485090.096987-5.6554710.0000R-squared0.448053Mean dep endent var-0.004983Adjusted R-squared0.433901S.D. dependent var0.644876S.E. of regression0.485202Akaike info criterion1.427829Sum squared resid18.36280Schwarz

21、criterion1.516512Log likelihood-54.82707Hannan-Quinn criter.1.463410F-statistic31.65901Durbin-Watson stat2.348799P rob(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( 12)的P值均小于0.05，参数显著

22、，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。F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： 200BM01 2014109Included'Observations:'S'1Q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Q-slatislic prot>abilili«5 adj u sle d for 2 ARMA t

23、erms)Auto CO rrelationP artial CorrelationACPACQ-EtatI匚1IE11<0181-0.181Z7560|11=12-0374-0.42014.550111匚130075-0.12215.1400.00011 140.124-0.05616.4S20X00'匚1'匚15-0.137-0.15918 1450.000111 1&-0014-0.06418.1620.0011111701S30.09121 2020.001111''8'-00300.03021 2830.002IE1111g-0.

24、145-0.03223 2460.0021111liOi0 0500.00723 4340.0031''1110.047-0.025237000.005C1匚112472-OJSO26.5S30,0031J111130.0S5O.OOD27.2030.0041 111U0.1550.03629 7130.0031匚11 115-0.160-o.og232 0070.0021111IE0 0050.0S232 0120.0041J111170.0830.02032.7320.00511：11 1TS0 094-0.07333 S63o.ooe1 111T9i*0 055-0.01

25、6339930.008111匚120'0015-0.13034 0160.013111 11210.091-0.031U.9500X14111 12200790.13635.6540.0171r 111230 0330.1181357770023匚1匚124-0 259-0.2163 S430.00411 111250 0310.01843 7570.006111匚126,0017-0.162437920.008111匸1270 022-0.12143 3510.011匸1匚1£3-0.102-0.21345.ie50.0111'11>290dS40J9949.

26、5210,00511 11 1300 030-0.05640 S360.007匸11匸131-0 255-0.1 u58 44S0.00111'1320 004-0.04658 4470.C011n 11 r-|<1 tCcnit匚 ft nn dn tfin-i由表3.6可知，由足白噪声的假设。将八个模型的估计结果进行汇总如下: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SARMA(1,1)( 1,0)121.2061811.2061

27、81是是是表3.7不同SARMA模型的特征汇总表SARMA(1,1)( 1,1)120.8624961.010301是是是SARMA(2,0) (1,0)121.0103011.424354是是否SARMA(2,0) (1,1)121.0002481.149124是是否SARMA(3,0) (1,0)121.2417641.391729是是是SARMA(3,0) (1,1)121.3917290.959325是是是综合来看，根据信息准则，应选择为：表3.8SARMA(1,1)( 1,1) 12对数据进行拟合是最优的。拟合结果SARMA(1,1) (1,1) 12模型估计结果Dep endent

28、 Variable: SDLMMethod: Least SquaresDate: 11/02/14 Time: 23:16Samp le (adjusted): 2008M01 2014M09Included observations: 81 after adjustmentsConvergence achieved after 13 iterationsMA Backcast: 2006M12 2007M12VariableCoefficientStd. Errort-StatisticP rob.C-0.0068210.002943-2.3177820.0232AR(1)0.018663

29、0.1411680.1322030.8952SAR(12)-0.2016230.120638-1.6713130.0988MA(1)-0.8339470.080352-10.378650.0000SMA(12)-0.8603910.041002-20.984270.0000R-squared0.701510Mean dep endent var-0.004983Adjusted R-squared0.685800S.D. dependent var0.644876S.E. of regression0.361475Akaike info criterion0.862496Sum squared

30、 resid9.930500Schwarz criterion1.010301Log likelihood-29.93107Hannan-Quinn criter.0.921797F-statistic44.65381Durbin-Watson stat2.003373P rob(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

31、.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 值，并将结果保存在 sdlnmf中，预测情况如下:图中左边是预测值与置信区间，右边是预测的误差。Foceaai: SDLHF Actual: SDLMForecaal sample: 2014M05 2014M09 Included observations: 5 Root Mean Squarw

32、i Error Mean Absolute ErrorMean At)5. Percenl ErrorTheil Inequality Coefficient0.648539 0 461327 62 91646 0.538154 0 0001070 35057Theil 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Blas ProportKjn Variance Proportion Covariance Proportton测均值与真实值偏离较小；variance proportion表

33、示方差误，用来反映预测波动与真实波动之间的差异，本例varianee proportion为0.649319，则说明预测波动与真实波动的差异较大;covarianee 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,因此预测效果较差。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,附录具体数据表5.1社会融资规模M指标社会融资规模2002-0815852003-071344地区全国2002-0935072003-083321频度月2002-107952003-094040单位亿元2002-1118052003-1012182002-01-4

34、722002-1231092003-1118322002-022892003-0133862003-1224982002-0331362003-029982004-0121142002-0411512003-0340412004-024382002-0517742003-0426222004-0365572002-0626212003-0529712004-0427312002-078132003-0658422004-0524432004-0632292007-1242812011-06108732004-075902008-01108592011-0753932004-0815012008

35、-0247312011-08107412004-0929812008-0363912011-0942792004-104832008-0470762011-1079082004-1119772008-0556782011-1195812004-1235862008-0659762011-12127442005-0136202008-0748902012-0197542005-028242008-0845752012-02104312005-0341892008-0956592012-03187042005-0419992008-1012882012-0496372005-0519682008-

36、1145172012-05114322005-0647232008-1281642012-06178022005-076292009-01139902012-07105222005-0820972009-02111312012-08124752005-0960412009-03220112012-09164622005-10-9742009-0454522012-10129062005-1123682009-05149592012-11112252005-1225242009-06210672012-12162822006-0163232009-0773882013-01254462006-0

37、217372009-0876502013-02107052006-0374722009-09118712013-03255032006-0433252009-1059852013-04176292006-0537852009-1195012013-05118712006-0638432009-1281002013-06103752006-0722542010-01205502013-0781912006-0833622010-02108772013-08158412006-0930772010-03138302013-09141202006-108942010-04149192013-1086

38、452006-1127882010-05108052013-11123102006-1238372010-06101962013-12125322007-0169082010-0772022014-0126003.942007-0230832010-08106462014-029369.772007-0363112010-09112242014-0320934.492007-0461032010-1086082014-0415259.452007-0538242010-11105542014-0514013.272007-0670422010-12107802014-0619673.172007-0731002011-01175602014-072736.942007-0869612011-0264682014-089576.522007-0952902011-03182122014-0910522.062007-1036882011-04136732007-1130732011-0510854存在冋题本次应用AR