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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(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:tttyx(1)t的无条件方差是常数,但是其条件分布为21|(0,)tttN22211ttqtq(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。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()ttt2122112211var()[var()]()tttttqtqtqtqEEEEE推出1var()1tq,说明1(0,)1tqN3、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,)tNARCH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:拉格朗日乘数检验建立辅助回归方程222011ttqtqteeev此处e是回归残差。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。Eviews操作:①先实施多元线性回归②view/residual/Tests/ARCHLMTest§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。输入lsrc,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.ofregression0.017603Akaikeinfocriterion-5.483292Sumsquaredresid0.076851Schwarzcriterion-5.386081Loglikelihood706.1198Durbin-Watsonstat2.042568观察:说明q选取为3确实比较恰当。4、ARCH模型是对的吗?如果ARCH模型选取正确,即回归残差的条件方差是按规律变化的,那么标准化残差就会服从标准正态分布,即不会有ARCH效应了。为什么?请思考。对q为3的ARCH模型做LMtest,发现没有了ARCH效应。注意,虽然是同一个检验名称,但是ARCH过程后是对标准化残差进行检验。注意观察被解释变量或者依赖变量是什么?ARCHTest:F-statistic0.238360Probability0.992099Obs*R-squared2.470480Probability0.991299TestEquation:DependentVariable:STD_RESID^2Method:LeastSquaresDate:10/21/04Time:21:56Sample(adjusted):20102254Includedobservations:245afteradjustingendpointsVariableCoefficientStd.Errort-StatisticProb.C1.1023710.2649904.1600430.0000STD_RESID^2(-1)-0.0385450.065360-0.5897410.5559STD_RESID^2(-2)-0.0038040.065308-0.0582520.9536STD_RESID^2(-3)-0.0573130.065303-0.
本文标题:第7章ARCH模型和GARCH模型
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