金融时间序列分析 第三次作业.docx

3.4模型为AR(1)-GARCH(1,1)，假定t服从自由度为v的标准化的t分布，导出数据的条件对数似然函数。数据为r=r1, r2, rn模型为rt=u+1rt-1+atat=ttt2=0+1at-12+1t-12由于t服从自由度为v的标准化的t分布，所以有t的概率密度函数为f(t)=v+12v2v-21+t2v-2-(v+1)/2其中(x)为Gamma函数（x=0infyx-1e-ydy）由于at=tt，at的条件似然函数为f(am+1,at)=t=m+1Tv+12v2v-21t1+t2(v-2)t2-(v+1)/2所以对数条件似然函数为L=Tln(v+12)-ln(v2)-12ln(v-2)n- 12t=1Tln(t2)+(1+v)ln(1+ at2(v-2)t2)带入实际的数据T=t，at=rt-u-1rt-1，同时又有t2=0+1at-12+1t-12，所以有了第一个1后就可以递推出其余的t。3.5对Intel股票的对数收益率建立GARCH模型，并进行向前1到5步的波动率预测。数据的图形如下：同时ACF和PACF如下：可知模型的基本形式应该为MA(1)。尝试对残差建立ARMA(0,1)Garch(1,1)模型，结果为*-* GARCH Model Fit *-*Conditional Variance Dynamics -GARCH Model : sGARCH(1,1)Mean Model : ARFIMA(1,0,0)Distribution : norm Optimal Parameters- Estimate Std. Error t value Pr(|t|)mu 0.025807 0.006441 4.00645 0.000062ar1 0.027009 0.054726 0.49353 0.621640omega 0.001235 0.000615 2.00819 0.044624alpha1 0.089186 0.033309 2.67753 0.007417beta1 0.836646 0.055546 15.06232 0.000000LogLikelihood : 238.1461检验残差的ACF发现模型可以满足要求。所以最终拟合的Garch模型为(1-0.027009*B)yt=0.025807+tt =ut*hthtN(0,n2)ut2=0.001235+0.089186*at-12+0.836646*t-12下面是向前1到5步的预测结果*-* GARCH Model Forecast *-*Model: sGARCHHorizon: 10Roll Steps: 0Out of Sample: 00-roll forecast: sigma series3730.1233 0.024263740.1236 0.02603375 0.1240 0.02607376 0.1243 0.02608377 0.1246 0.02608其中372就是03年12月的数据下图是预测结果趋势图 3.6(a)利用对数收益率和5%的显著性水平检验对数收益率中的相关性。观察对数收益率的ACF图形可以发现明显的一阶相关性。取12阶滞后的Ljung&Box检验的结果如下 Box-Ljung testdata: mrk X-squared = 24.3218, df = 12, p-value = 0.01838发现有显著的自相关性。尝试对序列建立ARMA(1,0)模型arima(x = mrk, order = c(1, 0, 0)Coefficients: ar1 intercept -0.0911 0.0121s.e. 0.0380 0.0024sigma2 estimated as 0.004746: log likelihood = 868.06, aic = -1730.13残差mrk$residuals=(1+0.0911*B)mrkt没有序列相关。(b)利用Ljung&Box统计量，在6以及12个间隔下验证序列的ARCH效应。令arch=mrk$residuals2，进行Ljung&Box检验间隔为6： Box-Ljung testdata: arch X-squared = 25.0263, df = 6, p-value = 0.0003376间隔为12： Box-Ljung testdata: arch X-squared = 35.2562, df = 12, p-value = 0.0004263在5%的显著性水平下无论是6还是12的间隔都是有显著的ARCH效应。(c)对数据识别一个ARCH模型，然后拟合*-* GARCH Model Fit *-*Conditional Variance Dynamics -GARCH Model : sGARCH(1,0)Mean Model : ARFIMA(1,0,0)Distribution : norm Optimal Parameters- Estimate Std. Error t value Pr(|t|)mu 0.012269 0.002411 5.0894 0.000000ar1 -0.080221 0.040635 -1.9742 0.048358omega 0.004433 0.000298 14.8977 0.000000alpha1 0.066349 0.043029 1.5420 0.123084模型形式为：(1+0.080221*B)mrkt=0.012269+tt =ut*hthtN(0,n2)ut2=0.004433+0.066349*at-12下面是拟合的残差图以及置信区间，发现拟合是有效的。3.7(a)利用Ljung&Box统计量，在6以及12个间隔下验证对数收益率的ARCH效应。为了检验3m数据的ARCH效应，将Ljung&Box应用于mmm2序列6个间隔： Box-Ljung testdata: mmm2 X-squared = 22.5644, df = 6, p-value = 0.000956312个间隔 Box-Ljung testdata: mmm2 X-squared = 29.9574, df = 12, p-value = 0.002834无论是6或者12个间隔，都是在5%的水平下有显著的ARCH效应的。(b)用收益率平方的PACF识别一个ARCH模型并拟合。PACF图形为序列平方项有2阶的偏自相关，所拟合的应该是一个ARCH(2)模型。*-* GARCH Model Fit *-*Conditional Variance Dynamics -GARCH Model : sGARCH(2,0)Mean Model : ARFIMA(0,0,0)Distribution : norm Optimal Parameters- Estimate Std. Error t value Pr(|t|)mu 0.012084 0.002294 5.2671 0.000000omega 0.003195 0.000273 11.6992 0.000000alpha1 0.094952 0.048639 1.9522 0.050916alpha2 0.134261 0.056703 2.3678 0.017894模型为：yt=0.012084+tt =ut*hthtN(0,n2)ut2=0.003195+0.094952*at-12+0.134261* at-22(c)利用前690个数据重新拟合，并对691到695的数据进行预测*-* GARCH Model Fit *-*Conditional Variance Dynamics -GARCH Model : sGARCH(2,0)Mean Model : ARFIMA(0,0,0)Distribution : norm Optimal Parameters- Estimate Std. Error t value Pr(|t|)mu 0.011836 0.002297 5.1527 0.000000omega 0.003169 0.000274 11.5532 0.000000alpha1 0.098426 0.049248 1.9986 0.045653alpha2 0.137802 0.057628 2.3912 0.016792yt=0.011836+tt =ut*hthtN(0,n2)ut2=0.003169+0.098426*at-12+0.137802* at-22预测结果为*-* GARCH Model Forecast *-*Model: sGARCHHorizon: 10Roll Steps: 0Out of Sample: 00-roll forecast: sigma series691 0.06068 0.01184692 0.06510 0.01184693 0.06398 0.01184694 0.06447 0.01184695 0.06436 0.01184下面是预测图：(d)对3M股票的对数收益率建立ARCH-M模型，并在5%的水平下检验风险溢价为0的假设。考虑到数据的PACF图形中有2阶的回归，建立ARCH(2)-M的模型：rt=u+c*t2+atat=ttt2=0+1at-12+1t-12拟合结果为：*-* GARCH Model Fit *-*Conditional Variance Dynamics -GARCH Model : sGARCH(2,0)Mean Model : ARFIMA(0,0,0)Distribution : norm Optimal Parameters- Estimate Std. Error t value Pr(|t|)mu 0.030460 0.021888 1.39164 0.164033archm -0.297533 0.352062 -0.84511 0.398046omega 0.003172 0.000273 11.60546 0.000000alpha1 0.093721 0.047570 1.97017 0.048819alpha2 0.141510 0.056890 2.48744 0.012867rt=0.030460-0.297533t2+atat=ttt2=0.003172+0.093721at-12+0.141510 at-22下面是拟合结果的检验图：可知拟合是有效的。同时关注系数archm的t值，可以发现对应的P值大于5%，所以系数是显著的，也就是说风险溢价为0的假设不成立。(e)利用前690个数据建立一个EGARCH模型，并进行向前1到5步的预测。由于数据的一阶自相关性，对数据建立AR(1)EGARCH(1,1)的模型。*-* GARCH Model Fit *-*Conditional Variance Dynamics -GARCH Model : eGARCH(1,1)Mean Model : ARFIMA(1,0,0)Distribution : norm Optimal Parameters- Estimate Std. Error t value Pr(|t|)mu 0.010652 0.002204 4.83244 0.000001ar1 -0.064397 0.040323 -1.59705 0.110254omega -0.929434 0.454867 -2.04331 0.041022alpha1 -0.018260 0.031722 -0.57563 0.564866beta1 0.831434 0.082353 10.09601 0.000000gamma1 0.198772 0.062164 3.19754 0.001386所以模型为：rt=0.010652-0.064397*rt-1+atat=ttln(t2)=0+ 11+0.018260*B*g(t-1)g(t)= 0.831434*t +0.198772*|t |-E(|t |)预测结果为：*-* GARCH Model Forecast *-*Model: eGARCHHorizon: 10Roll Steps: 0Out of Sample: 00-roll forecast: sigma series691 0.05690 0.005967692 0.05796 0.010954693 0.05886 0.010633694 0.05961 0.010653695 0.06025 0.010652走势图为：3.8(a)建立一个带高斯新息的GARCH模型并检验首先检查数据GM的ACF和PACF图形GM数据的均值模型为MA(1)。对建立的MA(1)模型的残差平方再次检验其ACF和PACF发现合适的模型为ARMA(0,1)GARCH(3,1)，对此进行拟合：*-* GARCH Model Fit *-*Conditional Variance Dynamics -GARCH Model : sGARCH(3,1)Mean Model : ARFIMA(0,0,1)Distribution : norm Optimal Parameters- Estimate Std. Error t value Pr(|t|)mu 0.011782 0.002862 4.117242 0.000038ma1 0.063162 0.043295 1.458858 0.144604omega 0.000157 0.000078 2.018931 0.043494alpha1 0.084301 0.061810 1.363884 0.172604alpha2 0.002563 0.062282 0.041150 0.967176alpha3 0.000000 0.001070 0.000205 0.999837beta1 0.884549 0.033406 26.478370 0.000000rt=0.011782+ at+0.063162* at-1at=ttt2=0.000157+0.084301*at-12 +0.884549t-12（其中alpha2和alpha3的系数为不显著的）拟合结果的图形为：拟合为有效的。(b)建立一个带高斯新息的GARCH-M模型并检验根据上面检测的结论，所建立的GARCH-M模型形式为ARMA(0,1)GARCH(1,1)-M*-* GARCH Model Fit *-*Conditional Variance Dynamics -GARCH Model : sGARCH(1,1)Mean Model : ARFIMA(0,0,1)Distribution : norm Optimal Parameters- Estimate Std. Error t value Pr(|t|)mu 0.006953 0.013328 0.52168 0.601892ma1 0.061927 0.043448 1.42532 0.154063archm 0.079453 0.208605 0.38088 0.703292omega 0.000151 0.000073 2.05206 0.040164alpha1 0.085087 0.024677 3.44795 0.000565beta1 0.887439 0.031145 28.49382 0.000000所以拟合的模型为：rt=0.006953+0.079453*t2+at +0.061927*at-1at=ttt2=0.000151+0.085087at-12+0.887439t-12(c)建立一个带学生t新息的GARCH模型，估计自由度v并写出拟合的模型。在5%的水平下检验假设H0：v=6。对GM数据建立MA(1)GARCH(1,1)模型，同时将残差序列设为student分布。*-* GARCH Model Fit *-*Conditional Variance Dynamics -GARCH Model : sGARCH(1,1)Mean Model : ARFIMA(0,0,1)Distribution : std Optimal Parameters- Estimate Std. Error t value Pr(|t|)mu 0.010291 0.002590 3.9734 0.000071ma1 0.052045 0.042122 1.2356 0.216615omega 0.000141 0.000093 1.5105 0.130914alpha1 0.066805 0.025940 2.5754 0.010012beta1 0.906084 0.039036 23.2117 0.000000shape 9.100371 2.988367 3.0453 0.002325模型形式为rt=0.010291+ at+0.052045* at-1at=ttt2=0.000141+0.066805*at-12 +0.906084t-12同时可以看到，自由度的预估值为9.1，均方误差为2.988367，显著不为6。(d)对GM数据建立EGARCH模型。对GM数据建立ARMA(0,1)EGARCH(1,1)模型。*-* GARCH Model Fit *-*Conditional Variance Dynamics -GARCH Model : eGARCH(1,1)Mean Model : ARFIMA(0,0,1)Distribution : norm Optimal Parameters- Estimate Std. Error t value Pr(|t|)mu 0.009111 0.002624 3.4726 0.000515ma1 0.081075 0.042053 1.9279 0.053866omega -0.158949 0.088744 -1.7911 0.073278alpha1 -0.031275 0.025569 -1.2232 0.221263beta1 0.969952 0.016282 59.5731 0.000000gamma1 0.181106 0.045228 4.0043 0.000062rt=0.009111+at-0.081075* at -1at=ttln(t2)= -0.158949+ 11+0.031275*B*g(t-1)g(t)= 0.969952*t +0.181106*|t |-E(|t |)(e)对所有的模型进行向前1到6步的预测，并进行比较。这是EGARCH的预测结果：*-* GARCH Model Forecast *-*Model: eGARCHHorizon: 6Roll Steps: 0Out of Sample: 00-roll forecast: sigma series649 0.08271 0.026335650 0.08234 0.009111651 0.08197 0.009111652 0.08162 0.009111653 0.08128 0.009111654 0.08095 0.009111这是GARCH的预测结果：*-* GARCH Model Forecast *-*Model: sGARCHHorizon: 6Roll Steps: 0Out of Sample: 00-roll forecast: sigma series649 0.09606 0.026985650 0.09551 0.009676651 0.09498 0.009676652 0.09445 0.009676653 0.09394 0.009676654 0.09344 0.009676这是残差服从t分布的GARCH的预测结果：*-* GARCH Model Forecast *-*Model: sGARCHHorizon: 5Roll Steps: 0Out of Sample: 00-roll forecast: sigma series649 0.09430 0.022158650 0.09371 0.009051651 0.09314 0.009051652 0.09258 0.009051653 0.09204 0.009051654 0.09150 0.009051对比如下：sigmaseriesGARCHGARCHtEGARCHGARCHGARCHtEGARCH6490.096060.094300.082710.0221580.0269850.0263356500.095510.093710.082340.0090510.0096760.0091116510.094980.093140.081970.0090510.0096760.0091116520.094450.092580.081620.0090510.0096760.0091116530.093940.092040.081280.0090510.0096760.0091116540.093440.091500.080950.0090510.0096760.009111GARCH和GARCHt在方差上的预测相近，但是均值预测不同，反而是EGARC