時(shí)間序列分析ARMA模型實(shí)驗(yàn)

1、基于ARM模型的社會(huì)融資規(guī)模增長(zhǎng)分析ARMA模型實(shí)驗(yàn)第一部分 實(shí)驗(yàn)分析目的及方法一般說(shuō)來(lái)，若時(shí)間序列滿足平穩(wěn)隨機(jī)過(guò)程的性質(zhì)，則可用經(jīng)典的ARM/模型進(jìn)行建模 和預(yù)則。但是,由于金融時(shí)間序列隨機(jī)波動(dòng)較大,很少滿足ARM/模型的適用條件,無(wú)法直接采用該模型進(jìn)行處理。通過(guò)對(duì)數(shù)化及差分處理后，將原本非平穩(wěn)的序列處理為近似平穩(wěn)的序列，可以采用ARMA模型進(jìn)行建模和分析。第二部分實(shí)驗(yàn)數(shù)據(jù)2.1數(shù)據(jù)來(lái)源數(shù)據(jù)來(lái)源于中經(jīng)網(wǎng)統(tǒng)計(jì)數(shù)據(jù)庫(kù)。具體數(shù)據(jù)見(jiàn)附錄表5.1 2.2所選數(shù)據(jù)變量社會(huì)融資規(guī)模指一定時(shí)期內(nèi)（每月、每季或每年）實(shí)體經(jīng)濟(jì)從金融體系獲得的全部 資金總額，為一增量概念，即期末余額減去期初余額的差額，或當(dāng)期發(fā)行

2、或發(fā)生額扣除 當(dāng)期兌付或償還額的差額。社會(huì)融資規(guī)模作為重要的宏觀監(jiān)測(cè)指標(biāo)，由實(shí)體經(jīng)濟(jì)需求所 決定，反映金融體系對(duì)實(shí)體經(jīng)濟(jì)的資金量支持。本實(shí)驗(yàn)擬選取2005年11月到2014年9月我國(guó)以月為單位的社會(huì)融資規(guī)模的數(shù)據(jù) 來(lái)構(gòu)建ARMA模型，并利用該模型進(jìn)行分析預(yù)測(cè)。第三部分ARMA模型構(gòu)建3.1判斷序列的平穩(wěn)性首先繪制出M的折線圖，結(jié)果如下圖:圖3.1社會(huì)融資規(guī)模 M曲線圖從圖中可以看出,社會(huì)融資規(guī)模M序列具有一定的趨勢(shì)性,由此可以初步判斷該序列是非平穩(wěn)的。此外,m在每年同時(shí)期出現(xiàn)相同的變動(dòng)趨勢(shì)，表明m還存在季節(jié)特征。下面對(duì)m的平穩(wěn)性和季節(jié)性進(jìn)行進(jìn)一步檢驗(yàn)。為了減少m的變動(dòng)趨勢(shì)以及異方差性，先對(duì)m進(jìn)

3、行對(duì)數(shù)化處理，記為Im,其時(shí)序圖如下:圖3.2 Im曲線圖對(duì)數(shù)化后的趨勢(shì)性減弱，但仍存在一定的趨勢(shì)性，下面觀察 表3.1 lm的自相關(guān)圖lm的自相關(guān)圖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顯著的, 束的增加慢慢衰減至0,由此可以看出該序列表現(xiàn)出一定的平穩(wěn)性。 由

7、于存在較弱的趨勢(shì)性且均值不為零，選擇存在趨勢(shì)項(xiàng)的形式，并根據(jù) 束，單位根檢驗(yàn)結(jié)果如下：上表可以看出，該Im序列的PACf只在滯后一期、ACF隨著滯后結(jié)進(jìn)一步進(jìn)行單位根檢驗(yàn)，AIC自動(dòng)選擇之后結(jié)表3.2單位根輸出結(jié)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.單位根統(tǒng)計(jì)量 ADF=-8.674646小于臨界值，且 P為0.0000，因此該序列不存在單位根,即該序列是平穩(wěn)序列。由于趨勢(shì)性會(huì)掩蓋季節(jié)性，從lm圖中可以看出，該序列有一定的季節(jié)性，為了分觀察dlm的自相關(guān)表:表3.3 dlm的自相關(guān)圖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的自相關(guān)圖可知，dim在滯后期為12、24、36等差的自相關(guān)系數(shù)均顯著異于零。因此該序列為以12為周期呈現(xiàn)季節(jié)性，而且季節(jié)自相關(guān)系數(shù)并沒(méi)有衰減至零，因此為了考慮這種季

13、節(jié)性，進(jìn)行季節(jié)性差分，得新變量sdim :觀察sdim的自相關(guān)圖:表3.4 sdim的自相關(guān)圖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在滯后期24之后的季節(jié) ACF和PACF已衰減至零，下面對(duì) sdim建立SARMA模型。3.2模型參數(shù)識(shí)別由表3.4 sdim的自相關(guān)圖的自相關(guān)圖可知，偏自相關(guān)系數(shù)在3階后

18、都落在兩倍標(biāo)準(zhǔn)差的范圍以內(nèi)，即不顯著異于零。自相關(guān)系數(shù)在1階和12階顯著異于零。因此 SARMA(P,q)模型中選擇P、q均不超過(guò)3。此外，由于高階移動(dòng)平均模型估計(jì)較為困難而且自回歸模型可以表示無(wú)窮階的移動(dòng)平均過(guò)程,因此Q盡可能取小。擬選擇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)12八個(gè)模型來(lái)擬合 sdlnm。3.3模型參數(shù)估計(jì)以SARMA(1,0

19、)( 1,0)12模型為例，分析該模型的估計(jì)及殘差的檢驗(yàn)，其他模型類(lèi)似?；貧w結(jié)果為:表3.5 SARMA(1,0)( 1,0)12模型估計(jì)結(jié)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)與 sar( 12)的P值均小于0.05，參數(shù)顯著

22、，可以通過(guò)檢驗(yàn)。該模型AIC為1.427829，SC值為1.516512?；貧w結(jié)果的最后一部分表示該模型滯后多項(xiàng)式的反特征根，小于1,因此該模型是平穩(wěn)的。F面對(duì)殘差進(jìn)行檢驗(yàn)。觀察殘差的自相關(guān)圖:表3.6 SARMA(1,0) (1,0)12模型的殘差檢驗(yàn)結(jié)果Date; 11/02/14 Time:Sample： 200BM01 2014109Included'Observations:'S'1Q統(tǒng)計(jì)量可知?dú)埐畲嬖谧韵嚓P(guān)性，P值遠(yuǎn)小于0.05，因此殘差不滿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，由足白噪聲的假設(shè)。將八個(gè)模型的估計(jì)結(jié)果進(jìn)行匯總?cè)缦?AICSC平穩(wěn)性可逆性殘差是否滿 足白噪聲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模型的特征匯總表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是是是綜合來(lái)看，根據(jù)信息準(zhǔn)則，應(yīng)選擇為：表3.8SARMA(1,1)( 1,1) 12對(duì)數(shù)據(jù)進(jìn)行擬合是最優(yōu)的。擬合結(jié)果SARMA(1,1) (1,1) 12模型估計(jì)結(jié)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模型預(yù)測(cè)在SARMA(1,1)( 1,1)12估計(jì)方程下選擇動(dòng)態(tài)估計(jì)，預(yù)測(cè)2014年10月至12月的序列 值，并將結(jié)果保存在 sdlnmf中，預(yù)測(cè)情況如下:圖中左邊是預(yù)測(cè)值與置信區(qū)間，右邊是預(yù)測(cè)的誤差。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 不等系數(shù)中 bias proportion表示偏誤，即預(yù)測(cè)均值與真實(shí)均值的偏離程度，本例中bias proportion 的值為 0.000107，預(yù)Blas ProportKjn Variance Proportion Covariance Proportton測(cè)均值與真實(shí)值偏離較??；variance proportion表

33、示方差誤，用來(lái)反映預(yù)測(cè)波動(dòng)與真實(shí)波動(dòng)之間的差異，本例varianee proportion為0.649319，則說(shuō)明預(yù)測(cè)波動(dòng)與真實(shí)波動(dòng)的差異較大;covarianee proportion表示協(xié)方差誤，反映殘存非系統(tǒng)性預(yù)測(cè)誤差,本例中該值為0.350574,因此預(yù)測(cè)效果較差。該誤差占比越大，預(yù)測(cè)效果越好。本例中的協(xié)方差誤要小于方差誤,附錄具體數(shù)據(jù)表5.1社會(huì)融資規(guī)模M指標(biāo)社會(huì)融資規(guī)模2002-0815852003-071344地區(qū)全國(guó)2002-0935072003-083321頻度月2002-107952003-094040單位億元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存在冋題本次應(yīng)用AR