上證指數(shù)收益率ARCH效應(yīng)分析

1、上證指數(shù)收益率arch效應(yīng)分析本文以上證指數(shù)為研究對象，選取了從2001年1月2日到2006年12月29 日一個(gè)時(shí)間窗口總共1444個(gè)收盤價(jià)P(i=1,2.1444 )，并用這1444個(gè) 收盤價(jià)計(jì)算 出對數(shù)收益率Log(sh/sh(-1)為樣本數(shù)據(jù)，利用Eviews軟件對上證指數(shù)收益率 ARCH效應(yīng)進(jìn)行分析。一、序列平穩(wěn)性檢驗(yàn)將收盤價(jià)對數(shù)處理化后的對數(shù)收益率導(dǎo)入Eviews，利用單位根檢驗(yàn)，經(jīng)處理后的數(shù)據(jù)如圖1所示。Hull Hypothesis： SER01 has a unit rootExogenous Constant, Linear TrendLaa Length: Q (Autom

2、atic ari AIC. MAXLAG=23)t-StatisticProb/Augmented Did eyAFullertest stetistic-37.Q5543Q.QQQQTest critical .allies:1% level-3 9644215% level 1QA level-3.41293Q -S.12S4S8MacKinnon (1996 one*sided pAaiues.圖1、上證對數(shù)收益率ADF檢驗(yàn)結(jié)果生成圖如圖可以看出，P值很小，且ADF統(tǒng)計(jì)值在1% 5%及10%勺顯著水平下，單 位根檢驗(yàn)的臨界值分別為-3.964421,-3.412930及-3.128458

3、,檢驗(yàn)統(tǒng)計(jì)量 值為- 37.06543且絕對值很大，遠(yuǎn)小于相應(yīng)的DW臨界值。從而拒絕f表明2001年1月2日到2006年12月29日的對數(shù)收益率為平衡時(shí)間序列 不存在單位 根，也可通過下面的時(shí)序圖看出。.12 -1 .08 -圖2、上證對數(shù)收益率時(shí)序圖由時(shí)間序列圖可以看出，在相當(dāng)長的時(shí)間內(nèi)，上證對數(shù)收益率波動都比較小， 可見序列是平穩(wěn)的。二、自相關(guān)性檢驗(yàn)自相關(guān)系數(shù)表示的是當(dāng)前值與滯后值的相關(guān)系數(shù)，偏自相關(guān)系數(shù)考慮了所有滯后值之后的預(yù)測能力而計(jì)算當(dāng)前和滯后序列的相關(guān)性。用EVIEW曲的VIEW-CORRELOGRA成自相關(guān)圖，滯后階數(shù)為25,通過自相關(guān)圖可以看出，上證收益率具有自相關(guān)性。Date

4、: 05 31 12 Time: 01:37Sample: 11443Included observations:AC FAC Q-Stat ProbAutooorreistio n Partial Correiatio ni111111D 1111111111111111111111 0.026 0.026 1.0019 0.3172 -0.017 -0.010 1.4226 0.49113 0.039 0.039 3.5690 0.312k14 0.017 0,015 3.9038 0.40B11牧 0.017 0.018 4.4205 049116 0.010 -0.012 4.5779

5、 0.599111 0.043 0.0437 2696 0.4011S 0 008 D.012 7.3612 0.49Bl19 0.008 0.011 7.4579 0.590I10 0.033 0.029 9.0612 0.526l111 0.025 0.024 9 9947 0.531112 0.034 0.031 11.637 0.4751113 0.011 0.009 11.&10 0 543114 0.033 0.028 13.356 0.49915 0.016 0.012 13.739 0 545116 0.012 *0.015 13.963 0.6011117 -0.003 0.

6、008 13.978 0.669iI18 0.011 O.OOS U.162 0.713i19 -0.014 -D.018 U.440 0.757i20 0.029 0.030 15.656 0.73BI21 H勺阪0.069 21.3=2 043GII22 0.029 D.032 22.509 0.424123 -0.000 -0.009 22.609 0 484ih24 0 046 0.05025.770 0.365125 -0.019 -0.029 26.309 0 391圖3、上證對數(shù)收益率相關(guān)圖三、模型選擇由模型定階可以發(fā)現(xiàn)，在ARMA( p,q)中，分別選取(p,q)為(1,1),

7、(2,2), ,(3,3),(3,4)幾個(gè)數(shù)據(jù)進(jìn)行模型估計(jì)，觀察各模型的P值和T 統(tǒng)計(jì)量。MA Backcast: 1VariableCoefficientStd. Errort-StatisticProb.AR(1)-0.7183020.236526-3.0368810.0024MA(1)0.7518340.2240613.3554940.0008MA Backcast: 1 2VariableCoefficientStd. Errort-StatisticProb.AR(2)-0.6717000.487731-1.3771930.1687MA(2)0.6580660.4953791.328

8、4100.1843MA Backcast: 1 3VariableCoefficientStd. Errort-StatisticProb.AR(3)-0.7773380.182647-4.2559690.0000MA(3)0.7842410.1806484.3412750.0000MA Backcast: 0 3VariableCoefficientStd. Errort-StatisticProb.AR(3)0.0379550.0264671.4340320.1518MA(4)0.0152130.0265000.5740800.5660通過上面的數(shù)據(jù)可以看出，在選取P=3, Q=3寸，所對

9、應(yīng)的P直最小，T統(tǒng)計(jì)量最大。運(yùn)用該ARMA(3,3)輸出結(jié)果如下Dependent ariable. SERJ1Method： Least SquaresDate： D5/D1/12 Time: 0149Sample ladjusied :1-43Included obsenatians: 1440 after adjustments Cori-erjence achieved after 22 iterations MA Back cast 1 3VariableCoefficientStd. ErrorVStalisticProb.AR-0.7773380JS2647B4.2559690.

10、0000MA0.7842410 13Q6484.3412750.0000R-squared0003481Mean dependent .ar0.000160Adjusted R-squared0.002768S.D.dependent var0.013602S.E. of regression0 013583Akalka info criterion-5.75B651Sum squared resid0.266295Sch%ccrirterion-5.751329Log livelihood4143.229Hannan-Qiiidn criter-5.755S18Durbin-Watson s

11、tat1 941483Inverted AR Roots,46+T60ii46-.80i-.92Averted MA Roots.46+.80I.46-80!-.92圖4 ARMA模型輸出結(jié)果圖結(jié)果圖形，可寫出輸出結(jié)果的表達(dá)式:Rt=0.7842 t_3-0.7773Rt_3+ 4R2=0.003481 DW=1.941483件 34)(-4.25)四、異方差性檢驗(yàn)赤池信息量準(zhǔn)則AIC建立的ARM模型，在P=3, Q=3時(shí)AIC值最小， 故確定ARM（3, 3）來描述上證指數(shù)收益率。對ARM（3, 3）模型的殘差 進(jìn)行滯后四期的ARCF異方差性檢驗(yàn)。Heteroskedasticit. Tes

12、t： ARCHF-statistic7.463369Prob. F(4.1431)0 0000Obs*R-squared2?Prob. Chi-Squaret4;D.DOOOTest Equation:Dependent adable: RESID EMethod: Least SquaresDate: 05/01/12 Time: 04:19Sample (adjusted): 8 1443Included observations: 1436 alter adjustm : entsVariableCoefficientStd. Errort-StatistlcProb.C0.000138

13、1.56E-058.B923530.0000RESID*2(-1)0.0448450 0264871.6930920.0907FSIDft2(-2)0.0690350.0264052.6144740.0090RESID*2(4)0.0907690.0264083.4371750.0005RESID 屹 HUQ.M75B40.0264391.796-3940.0726R-squared0.02043&Mean dependentvar0.000185Adjusted R-squared0.017699S.D. dependent1 皿0.000494S E of regression0.0004

14、B93ike infc criterion-12.40371Sum squared resid0.000343Schwarz criteri 口 門-1238537Lag lifeNhoodS910.366Hannan-Quinn criter.-12.39606F-statistic7.463369Durbin-/atson stat1.997750ProbT-statistic:0.000006圖5 ARCH效應(yīng)檢驗(yàn)結(jié)果輸出圖圖中F統(tǒng)計(jì)量為7.463369,對應(yīng)的概率趨于0,說明ARM (3, 3)模型顯著，觀察值氏為29.34558，對應(yīng)的概率趨于0,拒絕殘差不存在ARCH效應(yīng)的原假設(shè)，

15、說明上證指數(shù)綜合收益率存在明顯的ARCH效應(yīng)五、模型修正經(jīng)過上述檢驗(yàn)，可發(fā)現(xiàn)上證指數(shù)對數(shù)收益率且有自相關(guān)，異方差和平衡性的特點(diǎn)，故在ARM (3, 3)的基礎(chǔ)上加入GARC模型來擬合誤差效果較好，GARCH1, 1) 模型模擬效果較好，且描述異方差性簡潔，因此可采用ARMA(3,3)- GARCH( 1,1)模型來分析上證指數(shù)對數(shù)收益率Dependent Variable: SER01M&thod: ML-ARCH (Marquardt! - Normal distributionDate: 05/0112 Time: 04:36Sample (adjusted): 4 144$Include

16、d observations： 1440 after adjustmentsConverg 汕 oe achieved after 21 iterationsMA E ackcast 1 3Presample rsr1ance: backcast parameter = 0.7:GARCH = C(3J+C(4)-RESID(-1f2 + ClSrGARCHt-1)/ariabieCoefficientStd. Errorz-StatisticProb.-0 7630730.134349入5.5857410.00000.7809B50J2BBB56.0595420.0000.-ariance

17、Equationc8.67E-061.91 -064.S4B5440.0000RESIDL-1?-20.1210070.0135268.9459873.0000GARCHnl)0.8407580.01891944 4402C0.0000R-squared0.003239Mean dependent ,ar0.000160Adjusted R-squared0.002546S D.dependent var0.013602S.E of regression0.013504Akaike info criterion-5.881243Sum squared resid0265361Sdhwan en

18、terion.842936Loa likelihood4225.0 S5 Hannan-Quinn criter.-5.854409urbirbWatson stat1 940915Inserted AR Roots,46-.79i46+79i-91Inverted f.lL. Roots,46-.S0i4e-n.80i-.92圖6 GARCH(1,1)輸出結(jié)果圖將上述結(jié)果代入ARMA(3,3)- GARCH( 1, 1)可得Rt=0.781t_3-0.764Rt_3+ 桃(4.34)(-4.25)GARCH(1,1 方程是：a t斜6.00000867+0.121007+0 840578 a t-1t-1(4.548544)(8.945987)(44.44026)用EVIEWSfe成擬合殘差圖序列與實(shí)際圖，通過該圖可以分析得出，擬合效果較好，上證指數(shù)對數(shù)收益率服從ARMA(3,3)- GARCH( 1, 1)模型98，I8 v1 IIIi1 |rrrr