1、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
2、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 varAdjusted R-squared S.D. dependent var0.017364S.E. of regression Akaike
3、 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 Probability0.000001Obs*R-squared44.689540.000002Test Equation: RESID227Sample(adjusted): 20
4、10 2254 245 after adjusting endpoints0.0001105.34E-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)-0.0693880.069053-1.0
5、048540.3160RESID2(-7)0.0056170.0691780.0811930.9354RESID2(-8)0.1022380.0655451.5598060.1202RESID2(-9)0.0112240.0657850.1706190.8647RESID2(-10)0.0644150.0651570.9886130.32390.1824060.0003050.1474660.0006790.000627-11.868369.19E-05-11.711161464.875 F-statisticDurbin-Watson stat2.004802 Prob(F-statisti
6、c)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 ML - ARCH48Convergence achieved after 13 iterationsz-Statistic-0.0006400.000750-0.8528880.3937 Variance Equat
7、ion9.24E-051.66E-055.569337ARCH(1)0.2447930.0826402.9621420.0031ARCH(2)0.0814250.0774281.0516240.2930ARCH(3)0.4578830.1096984.174043-0.003823-0.0198840.017535-5.4959820.076872-5.426545705.73772.042013为了比较,观察将q放大对系数估计的影响54Convergence achieved after 16 iterations-0.0006010.000751-0.7999090.42389.38E-0
8、51.60E-055.8807410.2620090.0902562.9029590.00370.0419300.0705180.5945960.55210.4521870.1084884.168076ARCH(4)-0.0219200.050982-0.4299560.6672ARCH(5)0.0376200.0443940.8474080.3968-0.003550-0.0278300.017603-5.4832920.076851-5.386081706.11982.042568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
9、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 test,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?0.2383600.9920992.4704800.991299 STD_RESID2561.1023710.2649904.160043STD_RESID2(-1)-0.0385450.065360-0.5897410.5559STD_RESID2(-2)-0.0038040.065308-0.0582520.9536STD_RESID
10、2(-3)-0.0573130.065303-0.8776490.3810STD_RESID2(-4)-0.0103250.065277-0.1581690.8745STD_RESID2(-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)-0.0106530.065278-0.163
11、1940.8705STD_RESID2(-10)-0.0202110.065228-0.3098450.75700.0100841.007544-0.0322212.1127472.1465144.4094261078.1604.566625-529.15462.000071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/make 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
12、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 variance)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
13、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、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(Akai
14、ke 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(-1) INFLATION 11/19/04 Time: 10:37 1941 1985 45 after adjusting endpoints2.4328590.8163452.9801840.0047INFLATION(-1)0.4932130.131
15、1573.7604660.00050.2474774.7400000.2299764.1167843.6125195.450114561.16255.530410-120.627614.141101.6124420.000507检验ARCH效应0.2159500.9533081.2311920.94185046 1946 1985 40 after adjusting endpoints9.2705227.4255671.2484600.2204-0.0311620.170116-0.1831840.8557-0.0068860.170151-0.0404690.96800.1162610.1
16、695050.6858880.49740.0185450.1706200.1086940.91410.1279060.1686430.7584390.45340.03078012.28323-0.11175334.1508836.0085810.1428744085.0010.39620-196.85741.034796习题2:Lin的数据集 点击usinf文件series dp=100*D(log(p)ls dp c dp(-1) dp(-2) dp(-3) DP10 1951:1 1983:4 132 after adjusting endpoints0.1099070.0634051.7334100.0854DP(-1)0.3935830.0844324.661536DP(-2)0.2030930.0894522.2704050.0249DP(-3)0.3020730.0841853.5882140.6968251.0213730.6897190.7114120.3962771.01642820.100541.103785-63.0842398.065991.9709
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