第7章ARCH模型和GARCH模型经济学高等教育教育专区.docx

1、第7章ARCH模型和GARCH模型经济学高等教育教育专区第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

2、型的定义Engle（1982）提出ARCH模型（autoregressive 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) + E

3、YVar(X|Y)由于，所以条件均值为0，条件方差为。那么，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

4、序列。series r=log(p)-log(p(-1)点击序列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

5、 Mean dependent var0.000432Adjusted 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 Tes

6、t:F-statistic5.220573 Probability0.000001Obs*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. C0.000

7、1105.34E-052.0601380.0405RESID2(-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)0.0056170.0691780.081

8、1930.9354RESID2(-8)0.1022380.0655451.5598060.1202RESID2(-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 resi

9、d9.19E-05 Schwarz criterion-11.71116Log 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m

10、ethod中选取ARCH得到如下结果Dependent Variable: 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)0.244793

11、0.0826402.9621420.0031ARCH(2)0.0814250.0774281.0516240.2930ARCH(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-5.42

12、6545Log likelihood705.7377 Durbin-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

13、EquationC9.38E-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-0.003550 Mean dependent var0.000432Adjusted R-squared-0.027830 S

14、.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LM tes

15、t，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？ARCH Test:F-statistic0.238360 Probability0.992099Obs*R-squared2.470480 Probability0.991299Test Equation:Dependent Variable: STD_RESID2Method: Least SquaresDate: 10/21/04 Time: 21:56Sample(adjusted): 2010 2254Included observations: 2

16、45 after adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. C1.1023710.2649904.1600430.0000STD_RESID2(-1)-0.0385450.065360-0.5897410.5559STD_RESID2(-2)-0.0038040.065308-0.0582520.9536STD_RESID2(-3)-0.0573130.065303-0.8776490.3810STD_RESID2(-4)-0.0103250.065277-0.1581690.8745STD_RESID2(

17、-5)0.0035370.0652800.0541850.9568STD_RESID2(-6)-0.0074200.065274-0.1136700.9096STD_RESID2(-7)0.0633170.0652640.9701650.3330STD_RESID2(-8)-0.0121670.065293-0.1863400.8523STD_RESID2(-9)-0.0106530.065278-0.1631940.8705STD_RESID2(-10)-0.0202110.065228-0.3098450.7570R-squared0.010084 Mean dependent var1.

18、007544Adjusted R-squared-0.032221 S.D. dependent var2.112747S.E. of regression2.146514 Akaike info criterion4.409426Sum squared resid1078.160 Schwarz criterion4.566625Log likelihood-529.1546 F-statistic0.238360Durbin-Watson stat2.000071 Prob(F-statistic)0.992099方程整体是不显著的。还可以观察标准化残差ARCH建模以后，procs/mak

19、e residual series/可以产生残差和标准化残差，以分别下是残差和标准化残差。可以看出没有了集群现象。还可以观察波动率（条件方差）的图形。对比r和残差的图形，发现条件方差的起伏与波动率的大小一致。ARCH建模以后，procs/make garch variance series/ 得到结论：ARCH模型确实很好描述了股票市场收益率的波动性。可以观察系数之和小于1，满足平稳性条件。3、GARCH模型当q较大时，采用Bollerslov（1986）提出的GARCH模型（Generalized ARCH）1、模型定义条件方差方程 均值 ：过去的条件方差（也即预测方差，forecast v

20、ariance）注意：均值方程中若没有解释变量（即只有常数，如R C），则R2没有直观定义了，因此可为负）2、GARCH(p, q) 模型的稳定性条件计算扰动项的无条件方差：GARCH是协方差稳定的，因此是经典回归。3、GARCH模型的参数估计采用极大似然估计GARCH模型的参数。下面以GARCH(1, 1)为例。由GARCH(1, 1)模型可以得到yt的分布为由正态分布的定义公式，得到yt的pdf为第t个观察样本的对数似然函数值为其中注意yi和yj之间不相关，因而是独立的。似然函数为取对数就得到了所有样本的对数似然函数。其中条件方差项以非线性方式进入似然函数，所以不得不使用迭代算法求解。4、

21、模型的选择两条原则：1） 若ARCH(q)中q太大，比如q大于7时，则选择GARCH(p, q)2） 使用AIC和SC准则，选择最优的GARCH模型3） 对于金融时间序列，一般选择GARCH(1, 1)就够了。回顾AIC和SC定义：1）AIC准则（Akaike information criterion）AIC越小越好，结合如下两者：K（自变量个数）减少，模型简洁LnL增加，模型精确2）SC准则（Schwaz criterion）习题1：通货膨胀率有ARCH效应吗？Greene P572点击数据文件usinf_greene_p572。进行回归ls inflation c inflation(-

22、1)Dependent Variable: INFLATIONMethod: Least SquaresDate: 11/19/04 Time: 10:37Sample(adjusted): 1941 1985Included observations: 45 after adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. C2.4328590.8163452.9801840.0047INFLATION(-1)0.4932130.1311573.7604660.0005R-squared0.247477 Mean d

23、ependent var4.740000Adjusted R-squared0.229976 S.D. dependent var4.116784S.E. of regression3.612519 Akaike info criterion5.450114Sum squared resid561.1625 Schwarz criterion5.530410Log likelihood-120.6276 F-statistic14.14110Durbin-Watson stat1.612442 Prob(F-statistic)0.000507检验ARCH效应ARCH Test:F-stati

24、stic0.215950 Probability0.953308Obs*R-squared1.231192 Probability0.941850Test Equation:Dependent Variable: RESID2Method: Least SquaresDate: 11/19/04 Time: 10:46Sample(adjusted): 1946 1985Included observations: 40 after adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. C9.2705227.42556

25、71.2484600.2204RESID2(-1)-0.0311620.170116-0.1831840.8557RESID2(-2)-0.0068860.170151-0.0404690.9680RESID2(-3)0.1162610.1695050.6858880.4974RESID2(-4)0.0185450.1706200.1086940.9141RESID2(-5)0.1279060.1686430.7584390.4534R-squared0.030780 Mean dependent var12.28323Adjusted R-squared-0.111753 S.D. depe

26、ndent var34.15088S.E. of regression36.00858 Akaike info criterion10.14287Sum squared resid44085.00 Schwarz criterion10.39620Log likelihood-196.8574 F-statistic0.215950Durbin-Watson stat1.034796 Prob(F-statistic)0.953308习题2：通货膨胀率有ARCH效应吗？Lin的数据集 点击usinf文件series dp=100*D(log(p)ls dp c dp(-1) dp(-2) dp

27、(-3)Dependent Variable: DPMethod: Least SquaresDate: 11/19/04 Time: 10:10Sample(adjusted): 1951:1 1983:4Included observations: 132 after adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. C0.1099070.0634051.7334100.0854DP(-1)0.3935830.0844324.6615360.0000DP(-2)0.2030930.0894522.2704050

28、.0249DP(-3)0.3020730.0841853.5882140.0005R-squared0.696825 Mean dependent var1.021373Adjusted R-squared0.689719 S.D. dependent var0.711412S.E. of regression0.396277 Akaike info criterion1.016428Sum squared resid20.10054 Schwarz criterion1.103785Log likelihood-63.08423 F-statistic98.06599Durbin-Watson