ARCH模型和GARCH模型及其matlab实现

第9章、ARCH模型和GARCH模型*****重要阅读材料：Engle,Robert,2004,riskandvolatility:econometricmodelsandfinancialpractice,AER,94(3):405-420.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（volatilityclustering）：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，-0.20.00.20.40.60.8500100015002000R§1、ARCH模型1、条件方差多元线性回归模型：tttyX条件方差或者波动率（Conditionvariance，volatility）定义为211var()var(|)ttttt其中1t是信息集。2、ARCH模型的定义Engle（1982）提出ARCH模型（autoregressiveconditionalheteroskedasticity，自回归条件异方差）。ARCH(q)模型：tttyx（1）t的无条件方差是常数，但是其条件分布为21|~(0,)tttN22211ttqtq（2）其中1t是信息集。方程（1）是均值方程（meanequation）2t：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（conditionalvarianceequation），由二项组成常数ARCH项2ti：滞后的残差平方习题：方程（2）给出了t的条件方差，请计算t的无条件方差。引理（方差分解公式）：【不作要求】Var(X)=Var[E(X|Y)]+E[Var(X|Y)]证明：（1）条件方差定义为Var(X|Y)=E{[X-E(X|Y)]2|Y}（2）注意，条件方差Var(X|Y)是随机变量Y的函数。展开，得到Var(X|Y)=E(X2|Y)－[E(X|Y)]2因此，E[Var(X|Y)]=E{E(X2|Y)－[E(X|Y)]2}=E{E(X2|Y)}－E{[E(X|Y)]2}=E(X2)－E{[E(X|Y)]2}（3）使用如下公式Var[g(Y)]=E{[g(Y)]2}-{E[g(Y)]}2，定义g(Y)=E(X|Y)并代入以上等式，得到Var[E(X|Y)]=E{[E(X|Y)]2}-{E[E(X|Y)]}2=E{[E(X|Y)]2}-{E(X)}2（3）因此，合并，得到E[Var(X|Y)]+Var[E(X|Y)]=E(X2)－E{[E(X|Y)]2}+E{[E(X|Y)]2}-{E(X)}2=E(X2)-{E(X)}2=Var(X)证明结束。习题的证明：利用方差分解公式：Var(X)=VarY[E(X|Y)]+EY[Var(X|Y)]由于21|(0,)tttN，所以条件均值为0，条件方差为2t。那么，21var()ttt2122112211111var()[var()]()var()var()var()var()?tttttqtqtqtqtqtqtqtEEEEE推出1var()1tq，说明1~(0,)1...tqN3、ARCH模型的平稳性条件在ARCH(1)模型中，观察参数的含义：当1时，var()t当0时，退化为传统情形，(0,)tNARCH模型的平稳性条件：1i（这样才得到有限的方差）4、ARCH效应检验ARCHLMTest：拉格朗日乘数检验针对ARCH模型22211ttqtq，建立辅助回归方程222011ttqtqteeev此处e是辅助回归方程的回归残差。原假设：H0：序列不存在ARCH效应即H0：120q可以证明：若H0为真，则22LM()mRq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。当然，还可以直接使用方差整体显著性检验（F检验：H0：除常数项外所有系数都是0）。Eviews操作：①先实施多元线性回归②view/residual/Tests/ARCHLMTest下面依据实例来学习ARCH模型。§2、GARCH模型的实证分析从收盘价，得到收益率数据序列。seriesr=log(p)-log(p(-1))点击序列p，然后view/linegraph0500100015002000500100015002000P-0.20.00.20.40.60.8500100015002000R1、检验是否有ARCH现象。首先回归。取2000到2254的样本（点击sample即可）。输入lsrc或者在quick中选择样本区间。得到-0.12-0.08-0.040.000.040.08200020502100215022002250RDependentVariable:RMethod:LeastSquaresDate:10/21/04Time:21:26Sample:20002254Includedobservations:255VariableCoefficientStd.Errort-StatisticProb.C0.0004320.0010870.3971300.6916R-squared0.000000Meandependentvar0.000432AdjustedR-squared0.000000S.D.dependentvar0.017364S.E.ofregression0.017364Akaikeinfocriterion-5.264978Sumsquaredresid0.076579Schwarzcriterion-5.251091Loglikelihood672.2847Durbin-Watsonstat2.049819问题：这样进行回归的含义是什么？其次，view/residualtests/ARCHLMtest，得到ARCHTest:F-statistic5.220573Probability0.000001Obs*R-squared44.68954Probability0.000002TestEquation:DependentVariable:RESID^2Method:LeastSquaresDate:10/21/04Time:21:27Sample(adjusted):20102254Includedobservations:245afteradjustingendpointsVariableCoefficientStd.Errort-StatisticProb.C0.0001105.34E-052.0601380.0405RESID^2(-1)0.1415490.0652372.1697760.0310RESID^2(-2)0.0550130.0658230.8357660.4041RESID^2(-3)0.3377880.0655685.1516970.0000RESID^2(-4)0.0261430.0691800.3778930.7059RESID^2(-5)-0.0411040.069052-0.5952600.5522RESID^2(-6)-0.0693880.069053-1.0048540.3160RESID^2(-7)0.0056170.0691780.0811930.9354RESID^2(-8)0.1022380.0655451.5598060.1202RESID^2(-9)0.0112240.0657850.1706190.8647RESID^2(-10)0.0644150.0651570.9886130.3239R-squared0.182406Meandependentvar0.000305AdjustedR-squared0.147466S.D.dependentvar0.000679S.E.ofregression0.000627Akaikeinfocriterion-11.86836Sumsquaredresid9.19E-05Schwarzcriterion-11.71116Loglikelihood1464.875F-statistic5.220573Durbin-Watsonstat2.004802Prob(F-statistic)0.000001得到什么结论？2、模型定阶：如何确定q实施ARCHLMtest时，取较大的q，观察滞后残差平方的t统计量的p－value即可。此处选取q＝3。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DependentVariable:RMethod:ML-ARCHDate:10/21/04Time:21:48Sample:20002254Includedobservations:255Convergenceachievedafter13iterationsCoefficientStd.Errorz-StatisticProb.C-0.0006400.000750-0.8528880.3937VarianceEquationC9.24E-051.66E-055.5693370.0000ARCH(1)0.2447930.0826402.9621420.0031ARCH(2)0.0814250.0774281.0516240.2930ARCH(3)0.4578830.1096984.1740430.0000R-squared-0.003823Meandependentvar0.000432AdjustedR-squared-0.019884S.D.dependentvar0.017364S.E.ofregression0.017535Akaikeinfocriterion-5.495982Sumsquaredresid0.076872Schwarzcriterion-5.426545Loglikelihood705.7377Durbin-Watsonstat2.042013为了比较，观察将q放大对系数估计的影响DependentVariable:RMethod:ML-ARCHDate:10/21/04Time:21:54Sample:20002254Includedobservations:255Convergenceachievedafter16iterationsCoefficientStd.Errorz-StatisticProb.C-0.0006010.000751-0.7999090.4238VarianceEquationC9.38E-051.60E-055.8807410.0000ARCH(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-0.003550Meandependentvar0.000432AdjustedR-squared-0.027830S.D.dependentvar0.017364S.E.ofregression0.017603Akaikeinfocr