自回歸條件異方差模型

1、ARCH&GARCH 的Stata 命令及實(shí)例下面以數(shù)據(jù)集“sp500.dta ”為例,對(duì)1981年1月至1991年4月的標(biāo)普股指的日收益率進(jìn)行ARCH/GARCH 分析。 首先看日對(duì)數(shù)收益率的時(shí)序圖:line r t 存在波動(dòng)率聚集。作為對(duì)照,先考慮一個(gè)自回歸模型。為此,使用IC 來確定自回歸模型的階數(shù)。將AR(p模型看做一維的VAR(p,則可以使用VAR 系列的命令: varsoc r, maxlag(8Exogenous: _cons Endogenous: r 8 8626.62 .9867 1 0.321 .000118 -6.2109 -6.20395 -6.19167 7

2、 8626.13 .46369 1 0.496 .000117 -6.21126 -6.20509 -6.19417 6 8625.9 .0624 1 0.803 .000117 -6.21182 -6.20642 -6.19686 5 8625.86 7.5913* 1 0.006 .000117* -6.21251* -6.20789* -6.1997 4 8622.07 3.6252 1 0.057 .000118 -6.2105 -6.20664 -6.19982 3 8620.26 1.4422 1 0.230 .000118 -6.20991 -6.20683 -6.20137 2

3、 8619.54 6.443 1 0.011 .000118 -6.21012 -6.2078 -6.20371 1 8616.31 8.0317 1 0.005 .000118 -6.20851 -6.20697 -6.20424* 0 8612.3 .000118 -6.20634 -6.20557 -6.2042 lag LL LR df p FPE AIC HQIC SBIC Sample: 9 - 2783 Number of obs = 2775 Selection-order criteria大多數(shù)準(zhǔn)則都表明應(yīng)該選擇AR(5模型:上表顯示,5階滯后的系數(shù)依然顯著不為0.下面檢驗(yàn)O

4、LS 殘差的平方是否存在條件異方差: . predict e1,res(5 missing values generated. g e2=e12(5 missing values generated. ac e2. estimates store OLS_cons .0004126 .0002058 2.01 0.045 9.27e-06 .000816L5. .0524083 .0189315 2.77 0.006 .0153032 .0895134L4. -.0389725 .0189512 -2.06 0.040 -.0761162 -.0018287L3. -.0191588 .018

5、9694 -1.01 0.313 -.0563381 .0180205L2. -.047366 .0189518 -2.50 0.012 -.0845108 -.0102211L1. .0562165 .0189456 2.97 0.003 .0190837 .0933492rrr Coef. Std. Err. z P>|z| 95% Conf. Intervalr 6 .010818 0.0098 27.49399 0.0000Equation Parms RMSE R-sq chi2 P>chi2Det(Sigma_ml = .0001168 SBIC = -6.200174

6、FPE = .0001173 HQIC = -6.208356Log likelihood = 8635.83 AIC = -6.212981Sample: 6 - 2783 Number of obs = 2,778Vector autoregression. var r,lags(1/5 A u t o c o r r e l a t i o n s o f e 2pac e2 . corrgram e2,lags(20-1 0 1 -1 0 1LAG AC PAC Q Prob>Q Autocorrelation Partial Autocor1 0.1279 0.1279 45.

7、48 0.00002 0.1134 0.0987 81.28 0.00003 0.0804 0.0562 99.291 0.00004 0.0179 -0.0088 100.18 0.00005 0.1041 0.0919 130.35 0.00006 0.0273 0.0007 132.42 0.00007 0.0113 -0.0111 132.77 0.00008 0.0486 0.0353 139.36 0.00009 0.0250 0.0155 141.11 0.000010 0.0107 -0.0107 141.43 0.000011 0.0096 -0.0012 141.68 0.

8、000012 0.0055 0.0035 141.77 0.000013 0.0098 0.0010 142.03 0.000014 0.0108 0.0043 142.36 0.000015 0.0175 0.0153 143.22 0.000016 0.0032 -0.0042 143.25 0.000017 0.0034 -0.0024 143.28 0.000018 0.0097 0.0073 143.54 0.000019 0.0178 0.0154 144.43 0.000020 0.0050 -0.0036 144.5 0.0000從以上結(jié)果可以看出無論是自相關(guān)圖、偏自相關(guān)圖還是

9、Q檢驗(yàn)都表明OLS殘差存在自相關(guān)。為此,考察ARCH(p模型。為了確定p,下面估計(jì)殘差平方的自回歸階數(shù):ARCH(3模型:所有的ARCH項(xiàng)均很顯著。 下面估計(jì)更簡(jiǎn)潔的GARCH(1,1:Exogenous: _consEndogenous: e24 15251.3 .21651 1 0.642 9.9e-07 -10.9923 -10.9884 -10.9816 3 15251.2 8.7769* 1 0.003 9.8e-07* -10.9929* -10.9898* -10.9844* 2 15246.8 27.155 1 0.000 9.9e-07 -10.9905 -10.9881 -

10、10.984 1 15233.2 45.742 1 0.000 1.0e-06 -10.9814 -10.9798 -10.9771 0 15210.3 1.0e-06 -10.9656 -10.9649 -10.9635 lag LL LR df p FPE AIC HQIC SBIC Sample: 10 - 2783 Number of obs = 2774 Selection-order criteria. varsoc e2. estimates store ARCH3_cons .0000687 1.67e-06 41.10 0.000 .0000654 .000072L3. .1

11、18723 .0192852 6.16 0.000 .0809247 .1565214L2. .0359253 .0150815 2.38 0.017 .006366 .0654845L1. .1606503 .0104934 15.31 0.000 .1400837 .1812169archARCH_cons .0005433 .0001832 2.97 0.003 .0001843 .0009023L5. -.0103408 .0149861 -0.69 0.490 -.0397131 .0190314L4. -.0320676 .0170775 -1.88 0.060 -.065539

12、.0014037L3. -.0311578 .0176172 -1.77 0.077 -.0656869 .0033713L2. -.0094867 .0181812 -0.52 0.602 -.0451213 .0261478L1. .0522458 .0228176 2.29 0.022 .0075241 .0969676rrr Coef. Std. Err. z P>|z| 95% Conf. IntervalOPGLog likelihood = 8947.899 Prob > chi2 = 0.0261 Distribution: Gaussian Wald chi2(5

13、 = 12.72 Sample: 6 - 2783 Number of obs = 2,778 ARCH family regression. arch r L.r L2.r L3.r L4.r L5.r,arch(1/3 nologARCH項(xiàng)和GARCH項(xiàng)都很顯著。 下面考慮EGARCH(1,1:. estimates store GARCH11_cons 4.94e-06 5.10e-07 9.69 0.000 3.94e-06 5.94e-06 L1. .8642666 .0084824 101.89 0.000 .8476413 .8808919 garchL1. .0893403 .

14、0033301 26.83 0.000 .0828133 .0958673 archARCH_cons .0005794 .0001715 3.38 0.001 .0002433 .0009155 L5. -.0011428 .019924 -0.06 0.954 -.0401931 .0379075 L4. -.0434336 .0208996 -2.08 0.038 -.084396 -.0024712 L3. -.0119301 .0212921 -0.56 0.575 -.0536618 .0298017 L2. -.0153192 .0226133 -0.68 0.498 -.059

15、6404 .0290021 L1. .0666244 .0231129 2.88 0.004 .0213239 .1119249 rrr Coef. Std. Err. z P>|z| 95% Conf. Interval OPGLog likelihood = 8999.049 Prob > chi2 = 0.0160 Distribution: Gaussian Wald chi2(5 = 13.94 Sample: 6 - 2783 Number of obs = 2,778 ARCH family regression. arch r L.r L2.r L3.r L4.r

16、L5.r,arch(1 garch(1 nolog. estimates store EGARCH11_cons -3.584498 .185258 -19.35 0.000 -3.947597 -3.221399L1. 710.7785 22.08814 32.18 0.000 667.4866 754.0705archL1. .6225027 .0197884 31.46 0.000 .5837181 .6612874egarchARCH_cons .000546 .0001822 3.00 0.003 .0001888 .0009032L5. -.007748 .016581 -0.47

17、 0.640 -.0402461 .0247502L4. -.0392389 .0176344 -2.23 0.026 -.0738017 -.004676L3. -.0112005 .0173151 -0.65 0.518 -.0451374 .0227365L2. -.0311847 .0186149 -1.68 0.094 -.0676693 .0052999L1. .0531843 .021325 2.49 0.013 .011388 .0949806rrr Coef. Std. Err. z P>|z| 95% Conf. IntervalOPGLog likelihood =

18、 8940.547 Prob > chi2 = 0.0182 Distribution: Gaussian Wald chi2(5 = 13.62 Sample: 6 - 2783 Number of obs = 2,778 ARCH family regression. arch r L.r L2.r L3.r L4.r L5.r,arch(1 egarch(1 nolog在 GARCH(1,1模型中加入一個(gè) TARCH 項(xiàng)： . arch r L.r L2.r L3.r L4.r L5.r,arch(1 garch(1 tarch(1 nolog ARCH family regres

19、sion Sample: 6 - 2783 Distribution: Gaussian Log likelihood = 9016.976 Number of obs Wald chi2(5 Prob > chi2 = = = 2,778 12.30 0.0309 r r r L1. L2. L3. L4. L5. _cons ARCH arch L1. tarch L1. garch L1. _cons Coef. OPG Std. Err. z P>|z| 95% Conf. Interval .06502 -.006217 -.0101253 -.0373281 .0001

20、177 .0003849 .0226963 .0223478 .0206436 .0200826 .0194354 .0001735 2.86 -0.28 -0.49 -1.86 0.01 2.22 0.004 0.781 0.624 0.063 0.995 0.027 .0205361 -.0500179 -.0505861 -.0766892 -.0379749 .0000448 .109504 .0375839 .0303354 .0020329 .0382103 .000725 .126726 .0049348 25.68 0.000 .117054 .136398 -.0915795

21、 .008728 -10.49 0.000 -.108686 -.074473 .868188 5.20e-06 .0094734 4.86e-07 91.65 10.70 0.000 0.000 .8496205 4.25e-06 .8867554 6.15e-06 . estimates store TARCH1 TARCH 項(xiàng)很顯著，存在不對(duì)稱效應(yīng)。 . 上述估計(jì)都是高斯的，但是股指收益率可能存在厚尾現(xiàn)象。 將日收益率的核密度圖與正態(tài)分布對(duì)比： kdensityr,normopt(lpattern("-" Kernel density estimate 40 Dens

22、ity 0 -.3 10 20 30 50 -.2 -.1 0 Daily returns on S&P stock index Kernel density estimate Normal density .1 kernel = epanechnikov, bandwidth = 0.0015 尖峰厚尾。尤其在分布的左端厚尾。 下面對(duì)擾動(dòng)項(xiàng)的正態(tài)性進(jìn)行嚴(yán)格的統(tǒng)計(jì)檢驗(yàn)： . qui var r,lags(1/5 . varnorm Jarque-Bera test Equation r ALL chi2 6.1e+05 6.1e+05 df 2 2 Prob > chi2 0.0

23、0000 0.00000 Skewness test Equation r ALL Skewness -3.4412 chi2 5482.932 5482.932 df 1 1 Prob > chi2 0.00000 0.00000 Kurtosis test Equation r ALL Kurtosis 75.047 chi2 6.0e+05 6.0e+05 df 1 1 Prob > chi2 0.00000 0.00000 結(jié)果強(qiáng)烈拒絕擾動(dòng)項(xiàng)服從正態(tài)分布的原假設(shè)。 為此使用 t 分布，重新擬合 GARCH(1,1: . arch r L.r L2.r L3.r L4.r L

24、5.r,arch(1 garch(1 dist(t nolog ARCH family regression Sample: 6 - 2783 Distribution: t Log likelihood = Number of obs Wald chi2(5 Prob > chi2 = = = 2,778 12.82 0.0251 9135.102 r r r L1. L2. L3. L4. L5. _cons ARCH arch L1. garch L1. _cons /lndfm2 df Coef. OPG Std. Err. z P>|z| 95% Conf. Interv

25、al .0487162 -.0194403 -.0278249 -.0263707 -.0080478 .0005129 .0188628 .0188372 .0184064 .0183287 .0179622 .0001579 2.58 -1.03 -1.51 -1.44 -0.45 3.25 0.010 0.302 0.131 0.150 0.654 0.001 .0117457 -.0563606 -.0639009 -.0622942 -.0432531 .0002035 .0856866 .0174801 .008251 .0095528 .0271576 .0008224 .035

26、4573 .0070274 5.05 0.000 .021684 .0492307 .9391239 2.21e-06 1.351631 5.863724 .0107548 5.67e-07 .1309664 .5060182 87.32 3.90 10.32 0.000 0.000 0.000 .9180449 1.10e-06 1.094942 4.989009 .9602029 3.32e-06 1.608321 6.994419 . estimates store GARCH11t 為了對(duì)比，將以上各種擬合的系數(shù)估計(jì)值列表： . estimates table OLS ARCH3 GARCH11 EGARCH11 GARCH11t TARCH1 Variable r r L1. L2. L3.