金融计量-(G)ARCH模型在金融大数据中地的应用

1、(G)ARCH 模型在金融数据中的 应用实验目的理解自回归异方差(ARCH模型的概念及建立的必要性和适用的场合。了解(G) ARCK型的各种不同类型，如GARCH-M1型，EGARCH型和TARCH 模型。掌握对(G)ARCH真型的识别、估计及如何运用 Eviews软件在实证研究 中实现。实验步骤( 一 ) 沪深股市收益率的波动性研究1. 描述性统计(1) 数据选取与导入本实验选取中国上海证券市场 A股成分指数上证180和深圳证券市场A股成分指数深证300作为研究对象。分别从财经网站上下载了2010年 5月 4号到 2016年 4月 19号这将近6年的上证180和深证 300的每日收盘价，共

2、1448个。 其中，上证180指数的日收盘价以下记为sh,深证300指数的日收盘价以下记为sz。将下载的数据导入Eviews。(2) 生成收益率的数据列在 Eviews 的命令窗口中输入“genr rh=log(sh/sh(-1) ”，生成上证180指数的日收益率序列，记为 rh； 输入 “ genr rz=log(sz/sz(-1) ”， 生成深证300指数的日收益率序列，记为rz 。(3) 观察收益率的描述性统计量利用 Eviews 作出的沪市收益率rh 的描述性统计量如图1 所示。图1沪市收益率rh的描述性统计量从上图可以看出，样本期内，沪市收益率的均值为0.00395%,标准差为1.6

3、669%,偏度为-0.668201，左偏峰度为7.316683 ,远高于正态分布的峰度值 3, 说明沪市收益率rh具有尖峰和厚尾特征。JB统计量为1231.139,说明在极小水 平下，沪市收益率rh显著异于正态分布。利用Eviews作出的深市收益率rz的描述性统计量如图2所示。Series: RZSample 11446口加erva帅鸣1447MsanD 0001MedanD00105JMaKinmn.典3 弱2Llinirnm-0 0E66S6Sid DevD017526Skewness-0.7B100/KurtosisU.0719557JafC|L.e Bera门口白9慎Protwo ii

4、 hCi(XX)。算图2深市收益率rz的描述性统计量从上图可以看出，样本期内，深市收益率的均值为0.0128%,标准差为1.7926%,偏度为-0.781007，左偏峰度为6.079557 ,远高于正态分布的峰度值 3, 说明深市收益率rz也具有尖峰和厚尾特征。JB统计量为718.8909,说明在极小 水平下，沪市收益率rz也显著异于正态分布。而且深市收益率的标准差略大于 沪市，说明深市的波动性更大。2 .平稳性检验利用Eviews软件对rh和rz进行平稳性检验。沪市收益率rh的ADF检验结 果如图3所示；深市收益率rz的ADF佥验结果如图4所示。Null Hypatnesis： RH has

5、 a unit rootExogerlOjS： ConstantLag Length: 0 i Automatic- based on SC, maxlas=l0)t-StatisticProb *Augmented Dicke/-Fullertest statistic-37.042590.0000Test critical values：1% level5% level 10% level-3 434664-2.363333-2.567773“MacKinnon (11996) one-sided p-valuet.Augrn ente d DickeFu 11 eTe 与t E qu a

6、ti cnDependent Variable; D(RH)Mpthod- I qhM SquarssDate: 05/11fl5 Ume: 20 54Sample (adjusted): 3 144BIncluded observations 144S after adju3tnnent&VariableCoefficientStd. Error 1-StatisticProb-0 9744620 026307-37 D42590.0000C354E-D50.00043&D.080S240935gR-squared0.487244Mean dependent var&

7、.1BE=06Adjusted R-squared0.48 S889S.D. dependent var0 023279S.E. of regression0 016675Akaike info criterion-5.348441Sum squared resid0 401506Schwarz criterion5 341143Log likelihood3S60 923Hannan-Quinn criter.-5345717F-statlstic1372.154Durbin-Wats on stat1 991214Pro bF-stati stlo)0 000000图3 rh 的ADF检验

8、结果Null H/pothesis: RZ has a unit rootExogenous： ConstantLag Length: 0 (Automatic - based on SIC. msxlag=lO)t-StatisticProb*Augmented Dicke/-Fuller test statistic-55.6856900000Test critical values：1% level5% level10% level-3.434654-2.363333-2.567773*MacKinnon i1 390) one-sided p-value3.AuffrneritedDi

9、ckey-FullerTestEquatiixiDependent Variable' D(RZ)Method' Least SquaresDate: 05/11/16 Time: 20 54Sample (adjusted): 3 1448Included obseivations' 1446 after adjustmentsVariableCoefficientStd. ErrorStatisticProb.RZ(-1) C-0.9359980.0001090.0262570.000471-35.635690.2311740.00000.8172squaredAd

10、justed R-squared S.E. of regression Sum squared resid Log likelihood F-statisticPro o(.F-s1ati stic)0 43623 0 468255 0 01789g Q4fi2578 3766 555 12734690 000000Mean dependent var S.D. dependent var Akaike info criterion Schwarz crilerion Hannan-Quinn enter. Durbin-Watson stal-1.20E-05 0 024545 -52068

11、53 -5 J 99555 -5.204129 1.9&7091图4 rz 的ADF检验结果从这两个ADF检验结果可以看出，rh和rz的ADF检验值都小于临界值，说 明沪市收益率和深市收益率都是平稳的。3 .均值方程的确定及残差序列自相关检验通过对收益率的自相关检验，可以发现沪市的收益率与其滞后 7阶存在显著 的自相关，而深市的收益率也与其滞后7阶存在显著的自相关，因此建立的均值 方程如下：(1) 对收益率做自回归利用LS普通最小二乘法对rh和rh(-7)做回归，回归结果如图5所示Dependentsanab回 RHMethod: Least SquaresDate: 05/14； 1

12、S Time: 1734Sample fadjusted): 9 144SIncluded obseR'atiQns： 140 after adjustmentsvariableCoerriaentStd. Error t-StatisticProb.C6.72E-050.。438 Q 1535290 3730RH(-7)0.0476700,0262351 324055Q.0C33R-squared0.0Q231QMean deperdent varb85E4J5Adjusted R-sqijarfld0.001616S Dvar0.016631S.E of regression0.0

13、1661BAra ike info criterion-5.355294Sum squar&d resid0.397107Schwarz criterion-5.347971Log likelihood385A812Hannan-Quinn criter.-5.352560Fstatistic3 329370Durbin-Wats on stat1.945953ProbiF'Statistic0 0632&0图5收益率rh的回归结果忽略常数项的不显著，rh的均值方程估计为再又t rz和rz(-7)做回归,回归结果如图6所示。Dependent'ariable:

14、RZMethod： Least SquaresDate; 05/14/16 Time： 17:37Sample (adjusted): 9 144BIncluded observations： 1440 after adjustmentsVariableCoefficientSt! Error t-StatisticProbCQ.00C1590.0004710 3380320.7349RZ(-7)0.0610250.0262362 3259630.0202R-squared0.003746Mean aepenaemvarQ.D00167Adjusted R-squared0.003Q55S D

15、. dependentvar0.017695S.E. of regressionQ.0T7B&7Akaike info criterion-&210315Sum squared residC45&Q61Schwarz criterion5.202993Log likelihood3753427Haman-Quinn enter巧.2075吃F-slatistic541C1D2Durbin-Watson stat1.875428Pro b(F-statistic0.020159图6收益率rz的回归结果同样忽略常数项的不显著，rz的均值方程估计为(2) 用Ljung-Box

16、 Q统计量对均值方程拟合后的残差及残差平方做自相关检验得到rh残差的自相关系数acf和pacf值，如图7所示Date: 0£/14yiS Time: 17:42Sample 9 1440Included observations: 1440Autocorrelation Partial Correlation AC PAC Q-Stat Prob11 0 027 0 027 1.0430 0 3061112 4037 -0.038 3.0003 02231II1II3 -0.0Q3 刃 006 3 0837 537911n4 0 077 0,075 11416 002211115 0

17、.015 0.010 11,739 0039口II口II6 -0.094 -0 090 24 539 0.0001II1II7 -0.004 0.003 24 615 00011111R 0 053 0 048 23 503 DOQfl11)19 0.037 0,030 31443 000011110 -0.Q25 -0.011 32342 0.0001|11 0.043 0.Q37 35000 0 000111112 0.021 0.007 35 625 0 0001113 0.Q7J 0,062 42.776 0 00011114 ,079 -0.073 51 344 QUO。111115

18、 0.004 0.023 51 894 0 0M图7 rh 残差的自相关系数 acf和pacf值偏自相关系数显示rh残差不存在显著的自相关。再得到rh残差平方的自相关系数acf和pacf值，如图8所示Date:Time： 1744Sample: 9 indue白0 observations: 1441?Au 忸 5rr 之的Panial Carrelatidi AC PAC Q-Stat ProbI11o.ieo0 ISO46.33S0000I1_l20 23/U 212128u.ouoIn130.2130 152193.620000In114a0 007242400.000I1150 15C

19、0 051275 170000In11600 023297.270000Ii1170 1交0.046323 520.000I1180.100 019339 170.0001iI100or0 01035i 25o.coo1i11100 12E0 060374 930.0001i11110.09-GO1B306.36o.coo1j11120 0920 017399 140.00011:130 1790 119445740.00011114o ooe0.011455 000.0001j1II150.084-0.015469.33,000图8 rh 残差平方的自相关系数 acf和pacf值偏自相关系数

20、显示rh残差平方存在显著的自相关。再做出rz残差和rz残差平方的自相关系数图，如图 9和图10所示Date： 05/5 Time: 1746Sample: 9 1446Included1440Autocorrelation Pallai Correlation AC PAC Q-Stat Prob111O.OB20.062559610.01 fl12-0.041-0.0457 99550.01Sf30.0150.0218.33470.040J40.0290.0259.53770.04950 0160.015g S3120 077II116-0.042-0.04312.5140.051117-0

21、.0040.00212.5410.0 S4111Q0.0440.0401S.3290.05311g0.0120.00715.5490.07711110-0.064-0.06121,5650.017111111D.OOSo.oi g21.bd80.027111112-0.002-0.01321 &S50.041u1130.0700.07328.8470.0071114-0.047-0.05232.0030.004I111150.0070.02332.0630.006图9 rz残差的自相关系数acf和pacf值Date: 05/14/16 Time: 17:47Sample: 9 144B

22、Included cbservations： 1440AC PAC ChSlat Prob I I n 1I- I- I 1 1 I- I- 1 I _1 iBl T0 2070.20761.E320.0000 2720.240168760,0000 2380.162260 870,0000 2090 101313g80,0000 2000.082372070.0000 1333.002397 51O.O'QO0 1213.00041388O.O'OO0 1473.053把口 3。O.O'OO0 1250.03747338O.O'QO0.17fiQ.094518

23、36O.O'OO0.146D.054550 170.0000.103&.01G565.570.0000.207D.10E62772O.O'OO0.190D.090680440.00。0.119-3.02270095O.O'OOO1Z1 Ji 1131415Autocorrelation Partial Correlation图10 rz 残差平方的自相关系数 acf和pacf值从图中可以得到与rh类似的结论，即rz的残差不存在显著的自相关，而残 差平方存在显著的自相关。(3) 对残差平方做线性图对rh进行回归后提取残差，生成残差平方序列 resl ；对rz进行回

24、归后提 取残差，生成残差平方序列res2。利用软件作出resl和res2的线形图，如图11 和图12所示。RIES2.m-i250WC7501000125。图12 rz 残差平方线性图由这两个图可以看出，e t2的波动具有明显的时间可变性和集簇性，比如在 500和1000附近比较小，也就是说适合用 GARCH模型来建模。(4) 对残差进行ARCH-LM Test对rh做回归之后的窗口中进行 ARCH-LMTest,选择一阶滞后，得到检验结 果如图13所示。同样步骤得到rz的检验结果，如图14所示。Heteroskeda Elicit/ Test .ARCHF-statistic4S.32412

25、 Prob F(1J437)0 0000Obs*R-squared46.81700 Prob. Chi-Square(1)0.0000Test Equation:Dependem '. anaDie: Kh3iup-2Method: Least SquaresDate： 05/15/1S Time： 09:42Sample fsdjusted): 1& 1443Included ctseR'atons' 1439 afler adjustmentsVariableCoefficient SW. Error t-Statistic Pncb.C0 0002261

26、94E-0511 652620.0000RESIh2(FQ1B03770 0259435 951555。0 0。F?-sqgrM0.8253dMean d叩日ndgnt var0.000276AtijJStedR-squared0 031B61SD dependentvar0.000695SE of regression0.000684Akaike info criterioin-11.73512Sum squared resid0.000673 Schwarz criterion-11 72730Log livelihood3445422Hannan-Quinn criter-11r7323

27、9F-statistic姐32412DurUrbWatson stat2.065934Prob(F-st3t»stic)O.OQOOOO图 13 rh ARCH-LM TestHeteroskedast city Test: ARCHF-statistic6413170 Prqb. F1r1437)0 0000Obs-squared6147729 Prob. Oi-Square(l)0 0000TestEquatian:Dependentk anaDle: RE3IL '2Metliod- Least SquaresOatK 05/15/16 Time： 09:43Sampl

28、e factjuseti/ 10 1443Included obs日zati口ns： 1439 after adjustmentsVariabisCoeiicierrt Std. Error t-Statistic Prob.C0.0302532 04E 0512.427040.0000RESILZ2I-1)0 2366940 02f8108 008 22 70.0000R-squared0.042722Mean impendent var0.000319Adjusted R-squared0.0420568.D dependentvar0.000722S.E ofreqres&on0

29、,030707 Akaike mfo criterion11 67097Sum squared resid0,0JD717Sctiwaiz criterion-1166364Log likelihood33阻259Hannan-Quinn criter-1166823F-statistic04.1317QDurbinWatson sUt2,0S0575Prob(F-statistic)O.OJOQOO图 14 rz ARCH-LM TestARCH-LMTest检验的原假设是残差中一直到第 q阶都没有ARCH®象。在这 里q=1.由检验结果可以看出，rh的F检验统计量和LM检验统计量

30、都大于临界值， 因此拒绝原假设，认为rh残差中，ARCHC应是显著的。对于rz来说也是这样， rz残差中的ARCHS应也显著。4. GARC凄模型建模(1) GARCH(1,1膜型估计结果对rh和rz分别进行GARCH(1,1建模。其均值方程形式为其中r表示rh和rz都可以。其条件方差方程为利用软件对rh进行估计，估计结果如图15所示Dependent anable: RHMethod: HL - ARCIH (Marcuardt: - Narrr al distributionDate: 05/15/1B Time: 0959Sample (adjusted): 9 144Sincluded

31、 observatons: 1440 after adjustm&ntsConvergenca achieved arer 10 iterationsPresample an a nee: baclccast parameter = 0.7)GARCH = Ci：3) + C(4fRESlD-1 /2 + C(5)*GARGH11)VariableCo efficientStd. Errorz-StatisticProbC9.57E-0500003760 2554530.7984RH-7)0.0559120.0256002.1772580.0295Variance EquationC3

32、.45E-05B.54E-Q74 0345070.0001RESICMV20.0530780.0067957 8114190.0000GAR CHg)0.9331&20.007601122.77630.0000squared0.002242Mean deperxlent var6 36E-05Adjusted R-squared0.001548S . dependent var0.01S631& E. of regression0.016616Akaike info criterion5.547177Sum squared re&id口397134Schwarz ent

33、erien-5.523870Log likelihood3998.S&8Hannan-Quinn criter5 540343Duroin-Watson stat1 945433图15 rh 的GARCH(1 1)模型估计结果由估计结果可以看出，估计的模型为止匕外，除常数项外其他各系数全部显著，说明 rh序列具有显著的波动集簇 性。而且ARCH®和GARCHH系数之和为0.986，小于1,也符合理论。因此对 rh建立的GARCH0 1)模型是平稳的，具条件方差表现出均值回复，即过去的波 动对未来的影响是逐渐衰减的。再又t rz进行建模，估计结果如图16所示Dependent

34、. anable R2Method: ML - ARCH (Marquardt; - Normal distributionDate: 05715/16 Tine: 10:07Sample (adjustedj: 9 1448included otserv'ations： 1440 after adjustmentsConvergence achieved after 15 iterationsPresample xwrinnc。： backcast paramstar - 0.7) GARCH = 03) + C(4)*RESID(-1 Y2 + C(5 尸 GARCH(-1)Var

35、iableCo efficientStd. Error-StatisticProbC0 0002060.0003940 5213740.6018RZ0,0675310 02691125094500.0121Variance EquationC3.20E-0e9.05E-073,5395100.0004RESIW0.0437090.0063917.068637。口。00GARCHf-1)0.9397470.007006134 13220.0000R-sqtiare<J0.003099Mean dependent war0.000107Adjusted R-squared0.003006SD

36、 dependent0017795S E of regression0.017656AKaike info criterion-5.414273Sum squaredssi。0.459034Schwarz criterion-5.395966Leg likelihood3903.277Hannan-Quinn enter-5.407439Durbin-Watson stat1.375473图16 rz 的GARCH(1 1)模型估计结果估计的模型为对rz的GARCHC1 1)模型的估计结果分析与rh类似，除常数项外其他各系 数全部显著，说明rz序列具有显著的波动集簇性。而且ARCHED GAR

37、CH?系数 之和为0.988,小于1,也符合理论。因此对rz建立的GARCH0 1)模型是平稳 的，具条件方差表现出均值回复，即过去的波动对未来的影响是逐渐衰减的。(2) GARCH-M(1,1)古计结果对rh进行GARCH-M(1,1模型估计，在ARCH-MK中选择方差，得到rh的 GARCH-M(1,1模型估计结果如图17所示。Dependent Variable: RHMethQtr ML - ARCH (Llarquarcif: - Normal distributionDate; 05/15/16 Time 10:13Sample (adjusted): 9 1446Included

38、 observations' 1440 after adjustmentsConvergence achieved after24 iterationsPresample variance backcast (parameter - 0.7)GARCH =C(5)*RE 引口11/2 +C(G*GARCH1)VariableCoefficientStd. Errorz-StatisticProbGARCH0.1965203.07135800S39740.9490C5.53E-050.0007120.077570,938。RH(-7)0D559630.02568321789700.029

39、3Variance EquationC345E'O69.53E-C74 0202610.0001RESID(-ir20.0531140.00684 57.7587450.0000GARCH(-1)0 93311ED.007&05122.70090.0000R-squared0 002154Mean dependent var6 86E-05Adjusted R-squared0 000765S.D. dependentvar0.016631S E of regressionD.0'16625Akaike info criterion-5545791Sum squar&a

40、mp;d resid0 3971B9Schwarz criieriork-5523623Log likelihood3998970Hannan*Quinn enter.-5537591Durbin-Watson stat1 945021图17 rh 的GARCH-M(1,1)模型估计结果rh并由估计结果可以看出，均值方程中的 GARC项的系数并不显著，说明 不适合用GARCH-M1型来进行估计。同样步骤得到rz的GARCH-M(1,1模型估计结果，如图18所示De pendent'/alable: RZMethod: ML - ACH (Marquard; - Normal distr

41、ibutionDate： 05/15/16 Time： 10:17Sample (adjusted): 9 1448Included obse-vatians. 1440 artei adjustmentsCon mergence achieved after 2S iterationsPresample variance： backcast (parameter = 0 7)GARCH = C(4： + C(5)*RESIDt-1f2 + C(&rGARCHf-1)VarigbeCoefficientStd. Errorz-StatisticProb,GARCH1,47 292728

42、78J4Q口.5117280 5033C-0,0001300.000752-0173f070,8际R在乃0.0632360.02687425409460.0111Variance EquationC3.25E-069.64E-073 3752820.0007RESIDfra0,04 »2230.001255.908799。口。QQGARCHi-1)0.93903dO.OOT123131 7540.0000R-squaredi0,002493Mean cependentvar0.000167Adjusted R-sqjared0.001105SD dependentvar0.01739

43、5S E. of regression0.017695Akaike info criterion-5.413035Sum squared residQ.45964QSch 即 3Pz criterion-5.391117Log likelihood3903.421Hannan-Quinn crit&r-5.404835Durbin-Wats on stat1.97 1 20Qrz的GARCH-M(1,1模型估计结果与rh类似，即均值方程中的GARC项的系 数并不显著，说明rz不适合用GARCH-彼型来进行估计。(二)股市收益波动非对称性的研究1. TARCH1型估计结果在Thresho

44、ld order中填入1,得到rh的TARCH(1 1)模型估计结果如图19Variable: RHMethod ML- W?CH (MarguardO - Mcrmai distributionDats 05/15J16 rime 10:23Sample laiimsted) 9 144gIncluded observations: U40 alter adjstmentsCsneroerce acnie(/ed after 15 iterationsPie&aiTDle variance: tacxusttparameter = 0.7)GARCH = Cf3)* C(4rRESI

45、O(-1 Y2 + C5FRESIDM-CfE)*GARCH(-nVariabeCoefficientSt。 Errori-StatisticProb.C9 6OE-Q500003632505815),3321RH-7)0.05688900255952.174362。颊Variance Equationc3 44E-063 53E-074.033967o oaoiRESIDZF20.053183100838g5.3395600.0000RESD(-1)'2RESn(-1)<0：)-0.0002730.009160-D.029764Q.97S3GARCH-10.9332330.00

46、7599122.SJ6S0.0000R-squaied0.002242Mean d&pendent var6.86E-05Adjusted R-squared0.001543S.D. dependent/ar3.01&531SE. of regjessicciO.01GS1BAkaike info crteror5545789SumsquaFed resid0.397134Schwarz ofterf on-5 E23320Log ikelihood3998 96BHannan-Quinn enter.5.537538Djrhn-pVstson stal1.945434图19

47、rh 的TARCH(1, 1)模型估计结果估计结果显示，RESID(-1)A2*(RESID(-1)<0)的系数估计值小于0,并且不 显著，说明在沪市中并不存在收益波动的非对称性。同样步骤得到rz的TARCH(1 1)模型估计结果如图20所示。Deoendent Variable： RZMethod, ML - ARCH (Uarqjarlt) - Norma distributionDale. 05/15/lfi Time 1020Sample (adjuftecn： 9144fiIndLJded osservstons： 1440 after a<liustTientsConv

48、erger ce adiievei after "6 iterationsPrfisample variant: Dackcastparameter = 0.7iGARCH _ 7(& + C(4yRE3ID+ C(E )*RE3ID-ir2J(RESID(-1 )*0)C(6)*CARCH(-1)VariableCoefficien:Std. Emz-S:atisticPrat).C0 35E 0 =0.0004090.2342D40 8382RZl-7)0 0722810 0263A327386650.0062Variance Equationc433E-CE1.06 4

49、064 1357&00,Q0D0RE.SIDf-120035025O.COB70140254720.0001RESIDM)l2T(RESID(-1)<0)0.0285400.4109232 6129*20.0090GaRCH(-1)£336600.0077341如71超0.0000R-SQuarea0003603叱口 rtependent var0 000167Adjust&d R'Squar？d0.00291 CS D dependentvar0.017695S.E. ofregession0.01786EAkaike info critelon-5.

50、415270Sum squa 印 resid045912E'53&3302Log likslihood30D4.0QEHannan Quinn critor.5 407060Durbin-Watson stat1875496图20 rz 的TARCH(1, 1)模型估计结果估计结果显示，RESID(-1)A2*(RESID(-1)<0)的系数估计值大于0,并且显 著，说明在深市中存在收益波动的非对称性，即坏消息引起的波动比同等大小的好消息引起的波动要大。2. EGARCH型估计结果对rh进行EGARCH(1 1)估计，其估计结果如图21所示。Dependentanable

51、: RHMethod: LIL - .ACH (Marquardt: - Normal distributionDate： 05/15/1 a Time 10:29Sample fadus：ed;: 9 1448Incl jded observations: 1440 altor adjustmentsConvergence schieved after 17 iterationsPre sample va nance: ba ckcast (param eter = 07)LOG(GARCH) = C(3) + CH)*AESfRESID(-iySQRT(GARCHf-1)n +C(5)*R

52、ESID(-1 咆 SQ RT(GARCH(-1» + C(6rL0G(GARCH(-1)VariableCoeUcientStd. ErrorStatisticPnobC9.3BE-050,000377024674。0 3036RH(-7Qg5P50.02491120675630 0337Variance EquationC-0.1997730.Q3C940-& 4563030 000054)0.1276400.0138209 2359540.0000C(5)-0.0370120.007402-0 947290.3435C(6)0.9373420.003301299 05920.0000R-squared0.002294