GARCH模型实验时间序列

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1、word金融时间序列分析探究中国A股市场收益率的波动情况基于GARCH模型第一部分 实验背景自1990年12月,我国建立了、证券交易所,20多年来,我国资本市场在拓宽融资渠道、促进资本形成、优化资源配置、分散市场风险方面发挥了不可替代的重要作用,有力推动了实体经济的发展,成为我国市场经济的重要组成部分。自1980年第一次股票发行算起,我国股票市场历经30多年,就当前的股票市场来看,股票市场的动荡和股票的突然疯涨等一系列现象和问题值得我们深入思考和深入研究。第二部分 实验分析目的及方法沪深300指数是在以上交所和深交所所有上市的股票中选取规模大流动性强的最具代表性的300家成分股作为编制对象，成

2、为沪深证券所联合开发的第一个反应A股市场整体走势的指数。沪深300指数作为我国股票市场具有代表性的且作为股指期货的标的指数，以沪深300指数作为研究对象可以使得检验结果更加具有真实性和完整性，较好的反应我国股票市场的基本状况。本文在检验沪深300指数2011年1月4日到2012年12月12日的日收益率的相关时间序列特征的基础上，对序列r建立条件异方差模型，并研究其收益波动率。第三部分 实验样本3.1数据来源数据来源于国数据库。3.2所选数据变量沪深300指数编制目标是反映中国证券市场股票价格变动的概貌和运行状况，并能够作为投资业绩的评价标准，为指数化投资和指数衍生产品创新提供基础条件。故本文选

3、择沪深300指数2011年1月4日到2012年12月12日的日收益率作为样本，探究中国股票市场收益率的波动情况。 第四部分 模型构建4.1 单位根检验观察R的图形，如下所示：图4.2 R的柱状统计图从沪深300指数收益率序列r的线性图中，可观察到对数收益率波动的“集群”现象：波动在一些时间段较小，在有的时间段较大。此外，由图形可知，序列R没有截距项且没有趋势，故选择第三种形式没有截距项且不存在趋势进行单位根检验，检验结果如下：表4.1 单位根检验结果Null Hypothesis: R has a unit rootExogenous: NoneLag Length: 0 (Automatic

4、 - based on SIC, maxlag=21)t-StatisticProb.*Augmented Dickey-Fuller test statistic-31.292060.0000Test critical values:1% level-2.5673835% level-1.94115510% level-1.616476*MacKinnon (1996) one-sided p-values.单位根统计量ADF=31.29206小于临界值，且P为0.0000，因此该序列不是单位根过程，即该序列是平稳序列。图4.2 R的正态分布检验由图可知，沪深300指数收益率序列均值为0.0

5、10480，标准差为1.292140，偏度为0.164917，大于0，说明序列分布有长的右拖尾。峰度为4.828012，高于正态分布的峰度值3，说明收益率序列具有尖峰和厚尾的特征。统计量为137.5854，P值为0.00000，拒绝该对数收益率序列服从正态分布的假设。其中右偏表明总体来说，近年比较大的收益大多为正；尖峰厚尾表明有很多样本值较大幅度偏离均值，即金融市场由于利多利空消息波动较为剧烈，经常大起大落，从而有很多比较大的正收益和负收益。4.2 检验ARCH效应首先观察r的自相关图，其结果如下：Date: 12/16/14 Time: 08:16Sample: 1 957Included

6、observations: 957AutocorrelationPartial CorrelationACPACQ-StatProb| | |1-0.011-0.0110.12440.724| | |20.0340.0341.25100.535| | |3-0.004-0.0041.27030.736| | |4-0.006-0.0081.30820.860| | |50.0.2.10910.834| | |6-0.039-0.0383.60350.730| | |70.0640.0617.57110.372| | |80.0130.0177.72480.461| | |90.0270.8.4

7、1670.493| | |100.0520.05211.0.352| | |110.0170.01911.3430.415| | |12-0.-0.13.3270.346| | |13-0.033-0.14.4050.346| | |140.0.15.6300.336| | |150.0060.00515.6610.405| | |16-0.008-0.01215.7230.472| | |170.0080.00515.7920.539| | |180.0390.03417.2740.504| | |19-0.003-0.00417.2810.571| | |20-0.-0.18.1120.5

8、80| | |21-0.020-0.18.5180.616| | |220.0120.01818.6520.667| | |23-0.050-0.21.0.576| | |240.004-0.00121.0960.633| | |250.0110.00621.2050.681| | |26-0.016-0.01521.4460.719| | |270.0.05023.7640.643| | |280.0500.26.2550.559| | |29-0.025-0.03326.8860.578*| | |30-0.-0.31.1450.408| | |31-0.0050.00431.1700.4

9、58| | |32-0.052-0.33.8480.378| | |330.0130.01334.0070.419| | |34-0.049-0.36.4010.358| | |35-0.025-0.37.0.376| | |360.0120.00637.1600.415图4.3 R的自相关图由自相关图可知，该序列不存在自相关性。因此对R进行常数回归。其回归结果如下：表4.2 回归结果Dependent Variable: RMethod: Least SquaresDate: 12/16/14 Time: 08:10Sample: 1 957Included observations: 95

10、7VariableCoefficientStd. Errort-StatisticProb.C0.0104800.0417690.2509050.8019R-squared0.000000Mean dependent var0.010480Adjusted R-squared0.000000S.D. dependent var1.292140S.E. of regression1.292140Akaike info criterion3.351521Sum squared resid1596.162Schwarz criterion3.356603Log likelihood-1602.703

11、Hannan-Quinn criter.3.353457Durbin-Watson stat2.020315由上表可知，对常数的回归结果并不显著。下面得到残差平方的自相关图：Date: 12/16/14 Time: 08:18Sample: 1 957Included observations: 957AutocorrelationPartial CorrelationACPACQ-StatProb| | |10.0500.0502.37710.123|* |* |20.1070.10513.3800.001| | |30.0200.13.7690.003| | |40.0.14.9580.0

12、05| | |50.0200.01415.3310.009| | |60.0.16.2710.012|* |* |70.0840.07823.0700.002| | |80.0150.00123.2780.003| | |90.0.02725.2120.003| | |100.0610.05428.8180.001| | |110.014-0.00328.9990.002| | |120.0390.02530.4920.002| | |130.0.04433.2610.002| | |140.003-0.01833.2680.003| | |15-0.001-0.01433.2690.004|

13、 | |16-0.003-0.01133.2780.007| | |170.0200.33.6570.009| | |180.0430.04135.4500.008| | |190.006-0.35.4900.012| | |200.0320.01436.4860.013| | |210.0540.05239.3340.009| | |22-0.-0.03939.8290.011| | |230.0140.00140.0120.015| | |24-0.047-0.42.2160.012| | |250.0.00342.3220.017| | |26-0.016-0.00942.5850.|

14、| |27-0.-0.03043.0140.026| | |280.0250.43.6420.030| | |29-0.-0.44.9790.030| | |300.0.01945.7970.032| | |310.0.46.3430.038| | |320.0320.02747.3390.040| | |33-0.038-0.48.7650.038| | |340.0190.49.1340.| | |350.0250.03049.7340.| | |360.0160.01849.9840.061图4.4 残差平方的自相关图由上图可知，残差平方序列在滞后三阶并不异于零，即存在自相关性，进一步进

15、行lm检验，这里选取滞后将阶数为3，检验结果如下：表4.3 ARCH效应检验结果Heteroskedasticity Test: ARCHF-statistic4.373176Prob. F(3,950)0.0046Obs*R-squared12.99530Prob. Chi-Square(3)0.0046 由上表可知，p值为0.0046，因此在1%的显著水平下是存在ARCH效应的。选择滞后阶数更高的进行检验,发现滞后4阶也满足在1%的显著水平下存在ARCH效应，再选取其他高阶进行检验，发现高阶残差平方项均不满足。4.3 模型的估计分别估计ARCH（2）、ARCH（1）和GARCH（1，1），

16、由于R不存在自相关性，而且对常数回归也不显著，因此不对均值方程进行设定，之设定方差方程。AECH（2）估计结果如下：表4.4 arch（2）模型的估计结果Dependent Variable: RMethod: ML - ARCH (Marquardt) - Normal distributionDate: 12/16/14 Time: 08:38Sample: 1 957Included observations: 957Convergence achieved after 8 iterationsPresample variance: backcast (parameter = 0.7)G

17、ARCH = C(1) + C(2)*RESID(-1)2 + C(3)*RESID(-2)2VariableCoefficientStd. Errorz-StatisticProb.Variance EquationC1.4099610.07656018.416520.0000RESID(-1)20.0475310.0214202.2190530.0265RESID(-2)20.1062840.0239774.4328490.0000R-squared-0.000066Mean dependent var0.010480Adjusted R-squared0.000979S.D. depen

18、dent var1.292140S.E. of regression1.291507Akaike info criterion3.336256Sum squared resid1596.268Schwarz criterion3.351503Log likelihood-1593.399Hannan-Quinn criter.3.342063Durbin-Watson stat2.020182可以看出，残差平方滞后项的系数在5%的显著水平下都显著，因此选择arch（2）合适，再选择ARCH(1)。表4.5 arch（1）模型的估计结果Dependent Variable: RMethod: M

19、L - ARCH (Marquardt) - Normal distributionDate: 12/16/14 Time: 08:40Sample: 1 957Included observations: 957Convergence achieved after 7 iterationsPresample variance: backcast (parameter = 0.7)GARCH = C(1) + C(2)*RESID(-1)2VariableCoefficientStd. Errorz-StatisticProb.Variance EquationC1.5948100.06252

20、025.508840.0000RESID(-1)20.0432670.0207012.0901310.0366R-squared-0.000066Mean dependent var0.010480Adjusted R-squared0.000979S.D. dependent var1.292140S.E. of regression1.291507Akaike info criterion3.350173Sum squared resid1596.268Schwarz criterion3.360337Log likelihood-1601.Hannan-Quinn criter.3.35

21、4044Durbin-Watson stat2.020182可以看出，残差平方滞后项的系数在5%的显著水平下显著，因此选择ARCH（1）合适。下面对GARCH（1，1）进行估计。表4.6 GARCH（1，1）模型的估计结果Dependent Variable: RMethod: ML - ARCH (Marquardt) - Normal distributionDate: 12/16/14 Time: 08:42Sample: 1 957Included observations: 957Convergence achieved after 9 iterationsPresample var

22、iance: backcast (parameter = 0.7)GARCH = C(1) + C(2)*RESID(-1)2 + C(3)*GARCH(-1)VariableCoefficientStd. Errorz-StatisticProb.Variance EquationC0.0463730.0223702.0730260.0382RESID(-1)20.0383960.0091944.1762960.0000GARCH(-1)0.9348960.01941048.165150.0000R-squared-0.000066Mean dependent var0.010480Adju

23、sted R-squared0.000979S.D. dependent var1.292140S.E. of regression1.291507Akaike info criterion3.326751Sum squared resid1596.268Schwarz criterion3.341998Log likelihood-1588.850Hannan-Quinn criter.3.332558Durbin-Watson stat2.020182 以上模型的系数均满足非负性，而且在5%的水平下显著。 4.4模型残差的检验下面进行残差的自相关性的检验，检验结果如下：Date: 12/1

24、6/14 Time: 08:50Sample: 1 957Included observations: 957AutocorrelationPartial CorrelationACPACQ-StatProb| | |10.0020.0020.00420.949| | |20.0200.0200.39500.821| | |3-0.006-0.0060.42600.935| | |4-0.011-0.0110.54150.969| | |50.0250.0251.14810.950| | |6-0.050-0.0503.57430.734| | |70.0620.0617.29700.399|

25、 | |80.0050.0077.32610.502| | |90.0.0207.79880.555| | |100.0500.04910.1920.424| | |110.0110.01410.3130.502| | |12-0.041-0.11.9260.452| | |13-0.038-0.13.3050.425| | |140.0390.03814.7610.395| | |150.0090.00814.8320.464图4.5 ARCH(2)模型残差项的自相关图Date: 12/16/14 Time: 08:51Sample: 1 957Included observations:

26、957AutocorrelationPartial CorrelationACPACQ-StatProb| | |1-0.004-0.0040.01900.890| | |20.0320.0321.01080.603| | |3-0.005-0.0051.03510.793| | |4-0.007-0.0091.08870.896| | |50.0.1.86690.867| | |6-0.039-0.0393.34970.764| | |70.0.0647.56140.373| | |80.0120.0157.70170.463| | |90.0.0258.50820.484| | |100.

27、0.05411.4800.321| | |110.0150.01711.6990.387| | |12-0.044-0.13.6200.326| | |13-0.036-0.03214.8600.316| | |140.0340.03416.0130.313| | |150.0050.00516.0400.379图4.6 ARCH(1)模型残差项的自相关图Date: 12/16/14 Time: 08:52Sample: 1 957Included observations: 957AutocorrelationPartial CorrelationACPACQ-StatProb| | |10

28、.0.0.08940.765| | |20.0360.0361.31900.517| | |3-0.001-0.0011.31960.724| | |4-0.000-0.0011.31960.858| | |50.0300.2.21290.819| | |6-0.-0.3.89170.691| | |70.0600.7.39280.389| | |80.0050.0067.41370.493| | |90.0270.8.09450.525| | |100.0600.11.6070.312| | |110.0140.01311.7860.380| | |12-0.044-0.05413.6300

29、.325| | |13-0.033-0.14.6930.327| | |140.0380.03816.0880.308| | |150.0040.00316.1000.375图4.7 GARCH(1,1)模型残差项的自相关图观察残差的自相关图，可以看出均不存在自相关性。下面观察残差平方的自相关图。Date: 12/16/14 Time: 08:53Sample: 1 957Included observations: 957AutocorrelationPartial CorrelationACPACQ-StatProb| | |1-0.-0.0.52670.468| | |2-0.001-0

30、.0020.52790.768| | |3-0.002-0.0020.53040.912| | |40.0020.0020.53330.970| | |50.0010.0010.53360.991| | |60.0250.0251.11770.981| | |70.0700.5.88080.554| | |80.0010.0045.88150.660| | |90.0.8.85050.451| | |100.0690.13.4890.| | |110.0070.01113.5330.260| | |120.0250.02614.1220.293| | |130.0300.14.9920.308

31、| | |140.0070.00415.0390.376| | |15-0.005-0.00715.0620.447图4.8 ARCH(2)模型残差平方的自相关图Date: 12/16/14 Time: 08:54Sample: 1 957Included observations: 957AutocorrelationPartial CorrelationACPACQ-StatProb| | |1-0.000-0.0000.00020.990|* |* |20.1090.10911.4110.003| | |30.0010.00111.4130.| | |40.0270.01512.1010

32、.017| | |50.0050.00512.0.033| | |60.0.12.8620.|* |* |70.0.20.1080.005| | |80.0.00520.2120.| | |90.0430.02521.9980.009| | |100.0.06225.9050.004| | |110.005-0.00525.9290.007| | |120.0400.02627.4540.007| | |130.0470.04329.6030.005| | |140.004-0.01329.6170.009| | |15-0.005-0.01729.6450.013图4.9 ARCH(1)模型

33、残差平方的自相关图Date: 12/16/14 Time: 08:55Sample: 1 957Included observations: 957AutocorrelationPartial CorrelationACPACQ-StatProb| | |1-0.-0.0.94300.332| | |20.0.0442.86530.239| | |3-0.-0.0273.68940.297| | |4-0.-0.0274.23290.375| | |5-0.017-0.0164.51820.477| | |6-0.002-0.0014.52190.606| | |70.0650.0658.59

34、970.283| | |8-0.013-0.0118.75980.363| | |90.0500.04311.1440.266| | |100.0.13.6960.187| | |11-0.020-0.01914.0.228| | |12-0.002-0.00414.1030.294| | |130.0160.14.3580.349| | |14-0.006-0.00614.3930.421| | |15-0.-0.15.2260.435图4.10 GARCH(1,1)模型残差平方的自相关图ARCH(2)模型和GARCH(1，1)模型残差平方序列不存在自相关性，而ARCH(1)模型残差平方序列

35、存在自相关性，故ARCH（1）模型不适合。下面进行正态性检验。图4.11 ARCH(2)模型的柱形统计图图4.12 GARCH(1,1)模型柱形统计图由以上结果可知，均不满足正态分布。再进行ARCH效应的检验。表4.7 ARCH（1）模型残差ARCH效应检验Heteroskedasticity Test: ARCHF-statistic0.173845Prob. F(3,950)0.9141Obs*R-squared0.523445Prob. Chi-Square(3)0.9137表4.8 GARCH（1,1）模型残差ARCH效应检验Heteroskedasticity Test: ARCHF-statistic1.154565Prob. F(3,950)0.3261Obs*R-squared3.465643Prob. Chi-Square(3)0.3252LM检验的P值均大于5%，故不存在ARCH效应。下面对三个模型进行比较。表4.9 不同模型结果对比ARCH(2)ARCH(1)GARCH(1,1)AIC3.3362563.3501733.326751SC3.3515033.3603373.341998残差检验无自相关性，无ARCH效应，不满足正态性存在自相关性，无ARCH效应，不满足正态性无自相关性，无A