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1、第7章、ARCH模型和GARCH模型研究内容：研究随时间而变化的风险。（回忆：Markowitz均值方差投资组合选择模型怎样度量资产的风险）本章模型与以前所学的异方差的不同之处：随机扰动项的无条件方差虽然是常数，但是条件方差是按规律变动的量。波动率的聚类性（volatility clustering）：一段时间内，随机扰动项的波动的幅度较大，而另外一定时间内，波动的幅度较小。如图，1、ARCH模型1、条件方差多元线性回归模型：条件方差或者波动率（Condition variance，volatility）定义为其中是信息集。2、ARCH模型的定义Engle（1982）提出ARCH模型（auto
2、regressive conditional heteroskedasticity，自回归条件异方差）。ARCH(q)模型： （1）的无条件方差是常数，但是其条件分布为 （2）其中是信息集。方程（1）是均值方程（mean equation） ：条件方差，含义是基于过去信息的一期预测方差方程（2）是条件方差方程（conditional variance equation），由二项组成 常数 ARCH项：滞后的残差平方习题： 方程（2）给出了的条件方差，请计算的无条件方差。证明：利用方差分解公式：Var(X) = VarYE(X|Y) + EYVar(X|Y)由于，所以条件均值为0，条件方差为。那
3、么，推出，说明3、ARCH模型的平稳性条件在ARCH(1)模型中，观察参数的含义：当时，当时，退化为传统情形，ARCH模型的平稳性条件：（这样才得到有限的方差）4、ARCH效应检验ARCH LM Test：拉格朗日乘数检验建立辅助回归方程此处是回归残差。原假设：H0：序列不存在ARCH效应即H0：可以证明：若H0为真，则此处，m为辅助回归方程的样本个数。R2为辅助回归方程的确定系数。Eviews操作：先实施多元线性回归view/residual/Tests/ARCH LM Test2、GARCH模型的实证分析从收盘价，得到收益率数据序列。series r=log(p)-log(p(-1)点击序
4、列p，然后view/line graph1、检验是否有ARCH现象。首先回归。取2000到2254的样本。输入ls r c，得到Dependent Variable: RMethod: Least SquaresDate: 10/21/04 Time: 21:26Sample: 2000 2254Included observations: 255VariableCoefficientStd. Errort-StatisticProb. C0.0004320.0010870.3971300.6916R-squared0.000000 Mean dependent var0.000432Adju
5、sted R-squared0.000000 S.D. dependent var0.017364S.E. of regression0.017364 Akaike info criterion-5.264978Sum squared resid0.076579 Schwarz criterion-5.251091Log likelihood672.2847 Durbin-Watson stat2.049819问题：这样进行回归的含义是什么？其次，view/residual tests/ARCH LM test，得到ARCH Test:F-statistic5.220573 Probabili
6、ty0.000001Obs*R-squared44.68954 Probability0.000002Test Equation:Dependent Variable: RESID2Method: Least SquaresDate: 10/21/04 Time: 21:27Sample(adjusted): 2010 2254Included observations: 245 after adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. C0.0001105.34E-052.0601380.0405RESID2
7、(-1)0.1415490.0652372.1697760.0310RESID2(-2)0.0550130.0658230.8357660.4041RESID2(-3)0.3377880.0655685.1516970.0000RESID2(-4)0.0261430.0691800.3778930.7059RESID2(-5)-0.0411040.069052-0.5952600.5522RESID2(-6)-0.0693880.069053-1.0048540.3160RESID2(-7)0.0056170.0691780.0811930.9354RESID2(-8)0.1022380.06
8、55451.5598060.1202RESID2(-9)0.0112240.0657850.1706190.8647RESID2(-10)0.0644150.0651570.9886130.3239R-squared0.182406 Mean dependent var0.000305Adjusted R-squared0.147466 S.D. dependent var0.000679S.E. of regression0.000627 Akaike info criterion-11.86836Sum squared resid9.19E-05 Schwarz criterion-11.
9、71116Log likelihood1464.875 F-statistic5.220573Durbin-Watson stat2.004802 Prob(F-statistic)0.000001得到什么结论？2、模型定阶：如何确定q实施ARCH LM test时，取较大的q，观察滞后残差平方的t统计量的pvalue即可。此处选取q3。因此，可以对残差建立ARCH(3)模型。3、ARCH模型的参数估计参数估计采用最大似然估计。具体方法在GARCH一节中讲解。如何实施ARCH过程：由于存在ARCH效应，所以点击estimate，在method中选取ARCH得到如下结果Dependent Var
10、iable: RMethod: ML - ARCHDate: 10/21/04 Time: 21:48Sample: 2000 2254Included observations: 255Convergence achieved after 13 iterationsCoefficientStd. Errorz-StatisticProb. C-0.0006400.000750-0.8528880.3937 Variance EquationC9.24E-051.66E-055.5693370.0000ARCH(1)0.2447930.0826402.9621420.0031ARCH(2)0.
11、0814250.0774281.0516240.2930ARCH(3)0.4578830.1096984.1740430.0000R-squared-0.003823 Mean dependent var0.000432Adjusted R-squared-0.019884 S.D. dependent var0.017364S.E. of regression0.017535 Akaike info criterion-5.495982Sum squared resid0.076872 Schwarz criterion-5.426545Log likelihood705.7377 Durb
12、in-Watson stat2.042013为了比较，观察将q放大对系数估计的影响Dependent Variable: RMethod: ML - ARCHDate: 10/21/04 Time: 21:54Sample: 2000 2254Included observations: 255Convergence achieved after 16 iterationsCoefficientStd. Errorz-StatisticProb. C-0.0006010.000751-0.7999090.4238 Variance EquationC9.38E-051.60E-055.8807
13、410.0000ARCH(1)0.2620090.0902562.9029590.0037ARCH(2)0.0419300.0705180.5945960.5521ARCH(3)0.4521870.1084884.1680760.0000ARCH(4)-0.0219200.050982-0.4299560.6672ARCH(5)0.0376200.0443940.8474080.3968R-squared-0.003550 Mean dependent var0.000432Adjusted R-squared-0.027830 S.D. dependent var0.017364S.E. of regression0.017603 Akaike info criterion-5.483292Sum squared resid0.076851 Schwarz criterion-5.386081Log likelihood706.1198 Durbin-Watson stat2.042568观察：说明q选
