chap4_3
qaj
2023-12-18
Intel公司股票收益率的波动率建模实例
继续使用Intel公司股票从1973-1到2009-12的月度对数收益率数据， 有444个观测值。 上节中的ARCH(1)模型在模型检验中有一些不足， 比如关于标准化残差平方的Ljung-Box检验的滞后15和滞后20检验是显著的。 尝试用GARCH(1,1)模型来改进。记\(r_t\)为对数收益率序列。
数据读入：
library(readr)
library(zoo)##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericlibrary(xts)
library(lubridate)##
## Attaching package: 'lubridate'## The following objects are masked from 'package:base':
##
## date, intersect, setdiff, uniond.intel <- read_table(
"D:/齐安静 教学/时间序列分析/北大/ftsdata/m-intcsp7309.txt",
col_types=cols(
.default=col_double(),
date=col_date("%Y%m%d")
))
xts.intel <- xts(
log(1 + d.intel[["intc"]]), d.intel[["date"]]
)
tclass(xts.intel) <- "yearmon"
ts.intel <- ts(c(coredata(xts.intel)), start=c(1973,1), frequency=12)
at <- ts.intel - mean(ts.intel)- 对数收益率的时间序列图：
plot(ts.intel, ylab="log return", main="Intel Stock Price Monthly Log Return")
使用条件正态分布
- 采用正态条件分布建立GARCH(1,1)模型：
library(fGarch, quietly = TRUE)## NOTE: Packages 'fBasics', 'timeDate', and 'timeSeries' are no longer
## attached to the search() path when 'fGarch' is attached.
##
## If needed attach them yourself in your R script by e.g.,
## require("timeSeries")mod1 <- garchFit(~ 1 + garch(1,1), data=ts.intel, trace=FALSE)summary(mod1)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~1 + garch(1, 1), data = ts.intel, trace = FALSE)
##
## Mean and Variance Equation:
## data ~ 1 + garch(1, 1)
## <environment: 0x00000192ba4c3b28>
## [data = ts.intel]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1
## 0.01126568 0.00091902 0.08643831 0.85258554
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.0112657 0.0053931 2.089 0.03672 *
## omega 0.0009190 0.0003888 2.364 0.01808 *
## alpha1 0.0864383 0.0265439 3.256 0.00113 **
## beta1 0.8525855 0.0394322 21.622 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 312.3307 normalized: 0.7034475
##
## Description:
## Tue Dec 19 17:59:09 2023 by user: qaj
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 174.9040137 0.000000e+00
## Shapiro-Wilk Test R W 0.9709615 1.030281e-07
## Ljung-Box Test R Q(10) 8.0168436 6.271916e-01
## Ljung-Box Test R Q(15) 15.5006010 4.159946e-01
## Ljung-Box Test R Q(20) 16.4154862 6.905368e-01
## Ljung-Box Test R^2 Q(10) 0.8746345 9.999072e-01
## Ljung-Box Test R^2 Q(15) 11.3593520 7.267295e-01
## Ljung-Box Test R^2 Q(20) 12.5599382 8.954573e-01
## LM Arch Test R TR^2 10.5140057 5.709617e-01
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -1.388877 -1.351978 -1.389037 -1.374326- 结果优于arch(1)模型
vola <- volatility(mod1)
plot(ts(vola, start=start(ts.intel), frequency=frequency(ts.intel)),
xlab="年", ylab="波动率")
abline(h=sd(ts.intel), col="green")
- 可以看出波动率在1973-1974石油危机期间和2000年的互联网泡沫期间最高。 波动率图形中绿色横线为样本标准差，值为：
sd(ts.intel)## [1] 0.1269101- 按模型计算的对数收益率无条件标准差为:
tmpx <- coef(mod1)
unname(sqrt(tmpx["omega"]/(1 - tmpx["alpha1"] - tmpx["beta1"])))## [1] 0.122767模型计算的无条件标准差与样本标准差很接近。
标准化残差的时间序列图：
##plot(mod1, which=9)
resi <- residuals(mod1, standardize=TRUE)
plot(ts(resi, start=start(ts.intel), frequency=frequency(ts.intel)),
xlab="年", ylab="标准化残差")
- 除了个别异常值，标准化残差看不出不符合独立同分布序列的特征。
标准化残差的ACF:
plot(mod1, which=10)
- \(\tilde a_t^2\)的ACF:
plot(mod1, which=11)
- 用\(\hat\mu + 2\hat\sigma_t\)可以作为\(r_t\)的近似95%置信区间， 将置信下限和置信下限分别延\(t\)轴方向连成曲线，得到如下图形
plot(mod1, which=3)
vola <- volatility(mod1)
hatmu <- coef(mod1)["mu"]
lb.intel <- hatmu - 2*vola
ub.intel <- hatmu + 2*vola
ylim <- range(c(ts.intel, lb.intel, ub.intel))
x.intel <- c(time(ts.intel))
plot(x.intel, c(ts.intel), type="l",
xlab="年", ylab="对数收益率")
lines(x.intel, c(lb.intel), col="red")
lines(x.intel, c(ub.intel), col="red")
使用条件t分布
- 应用条件t分布的新息，拟合模型：
library(fGarch, quietly = TRUE)
mod2 <- garchFit(~ 1 + garch(1,1), data=ts.intel,
cond.dist="std", trace=FALSE)summary(mod2)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~1 + garch(1, 1), data = ts.intel, cond.dist = "std",
## trace = FALSE)
##
## Mean and Variance Equation:
## data ~ 1 + garch(1, 1)
## <environment: 0x00000192b12d95d0>
## [data = ts.intel]
##
## Conditional Distribution:
## std
##
## Coefficient(s):
## mu omega alpha1 beta1 shape
## 0.0165076 0.0011576 0.1059029 0.8171298 6.7723926
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.0165076 0.0051031 3.235 0.001217 **
## omega 0.0011576 0.0005782 2.002 0.045287 *
## alpha1 0.1059029 0.0372046 2.846 0.004420 **
## beta1 0.8171298 0.0580150 14.085 < 2e-16 ***
## shape 6.7723926 1.8572665 3.646 0.000266 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 326.2264 normalized: 0.734744
##
## Description:
## Tue Dec 19 17:59:10 2023 by user: qaj
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 203.4935971 0.000000e+00
## Shapiro-Wilk Test R W 0.9687607 3.970512e-08
## Ljung-Box Test R Q(10) 7.8777964 6.407723e-01
## Ljung-Box Test R Q(15) 15.5522542 4.124162e-01
## Ljung-Box Test R Q(20) 16.5048039 6.848548e-01
## Ljung-Box Test R^2 Q(10) 1.0660526 9.997694e-01
## Ljung-Box Test R^2 Q(15) 11.4988561 7.164967e-01
## Ljung-Box Test R^2 Q(20) 12.6150651 8.932826e-01
## LM Arch Test R TR^2 10.8074911 5.454849e-01
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -1.446966 -1.400841 -1.447215 -1.428776- 模型(6.7)隐含的\(a_t\)的无条件标准差为
tmpx <- coef(mod2)
unname(sqrt(tmpx["omega"]/(1 - tmpx["alpha1"] - tmpx["beta1"])))## [1] 0.1226396- 标准化残差相对于标准化t分布的QQ图
plot(mod2, which=13)
使用条件有偏t分布
- 对数收益率序列\(r_t\)的样本偏度为
skewness.test <- function(x, na.rm=TRUE){
if(na.rm) x <- x[!is.na(x)]
z <- (x - mean(x))/sd(x)
n <- length(x)
sk <- n/(n-1)/(n-2) * sum(z^3)
t.sk <- sk / sqrt(6/n)
p.sk <- 2*(1 - pnorm(abs(t.sk)))
c(estimate=sk, statistic=t.sk, pvalue=p.sk)
}
skewness.test(as.vector(ts.intel))## estimate statistic pvalue
## -5.563719e-01 -4.786092e+00 1.700598e-06偏度估计为\(-0.56\)， 偏度为零的零假设的检验p值很小， 所以收益率分布是显著左偏（负偏）的。 这样， \(\varepsilon_t\)的分布最好采用左偏的分布。
在fGarch的garchFit()中指定cond.dist=“sstd”， 则条件分布为有偏的标准化t分布：
library(fGarch, quietly = TRUE)
mod3 <- garchFit(~ 1 + garch(1,1), data=ts.intel,
cond.dist="sstd", trace=FALSE)summary(mod3)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~1 + garch(1, 1), data = ts.intel, cond.dist = "sstd",
## trace = FALSE)
##
## Mean and Variance Equation:
## data ~ 1 + garch(1, 1)
## <environment: 0x00000192b66f3548>
## [data = ts.intel]
##
## Conditional Distribution:
## sstd
##
## Coefficient(s):
## mu omega alpha1 beta1 skew shape
## 0.0133343 0.0011621 0.1049294 0.8177869 0.8717222 7.2344212
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.0133343 0.0053430 2.496 0.012572 *
## omega 0.0011621 0.0005587 2.080 0.037520 *
## alpha1 0.1049294 0.0358862 2.924 0.003456 **
## beta1 0.8177869 0.0559866 14.607 < 2e-16 ***
## skew 0.8717222 0.0629130 13.856 < 2e-16 ***
## shape 7.2344212 2.1018126 3.442 0.000577 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 328.0995 normalized: 0.7389628
##
## Description:
## Tue Dec 19 17:59:10 2023 by user: qaj
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 195.2180903 0.000000e+00
## Shapiro-Wilk Test R W 0.9692506 4.892633e-08
## Ljung-Box Test R Q(10) 7.8821240 6.403498e-01
## Ljung-Box Test R Q(15) 15.6249576 4.074053e-01
## Ljung-Box Test R Q(20) 16.5774063 6.802192e-01
## Ljung-Box Test R^2 Q(10) 1.0784338 9.997569e-01
## Ljung-Box Test R^2 Q(15) 11.9515703 6.826911e-01
## Ljung-Box Test R^2 Q(20) 13.0379307 8.757507e-01
## LM Arch Test R TR^2 11.1882790 5.128558e-01
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -1.450899 -1.395550 -1.451257 -1.429071输出的检验结果表明模型充分。 AIC为\(-1.451\)， 相比对称的标准化t分布的\(-1.447\)， 和正态分布时的AIC为\(-1.389\)， 有偏t分布的AIC较优。
输出中,偏度参数\(\xi\)的检验是\(H_0: \xi=0\)的检验， 但是我们需要的是\(H_0: \xi=1\)的检验。 计算\(z\)检验如下：
z <- (0.8717220 - 1) / 0.0629129
pv <- 2*(1 - pnorm(abs(z)))
c(statistic=z, pvalue=pv)## statistic pvalue
## -2.03897770 0.04145225- 作标准化残差相对于\(t^*(0.87, 7.23)\)分布的QQ图：
plot(mod3, which=13)
讨论和比较
- 将三个模型估计的波动率绘制在同一坐标系中进行比较， 可以看到三个模型拟合的波动率基本相同：
x.intel <- as.vector(time(ts.intel))
vola1 <- volatility(mod1)
vola2 <- volatility(mod2)
vola3 <- volatility(mod3)
matplot(x.intel, cbind(vola1, vola2, vola3),
type="l",
lty=1, col=c("green", "blue", "red"),
xlab="年", ylab="波动率")
legend("top", lty=1, col=c("green", "blue", "red"),
legend=c("标准正态分布", "对称标准化t分布", "有偏标准化t分布"))
- 下面比较三个模型的波动率超前1到12步的预测结果， 预测基于截止到2009年12月的数据：
library(tibble)
p1 <- predict(mod1, n.ahead=12)[["standardDeviation"]]
p2 <- predict(mod2, n.ahead=12)[["standardDeviation"]]
p3 <- predict(mod3, n.ahead=12)[["standardDeviation"]]
pred.tab <- tibble(
"预测步数"=1:12,
"正态"=p1,
"对称t"=p2,
"有偏t"=p3
)
knitr::kable(pred.tab, digits=4)| 预测步数 | 正态 | 对称t | 有偏t |
|---|---|---|---|
| 1 | 0.0975 | 0.0951 | 0.0955 |
| 2 | 0.0993 | 0.0975 | 0.0979 |
| 3 | 0.1009 | 0.0997 | 0.1000 |
| 4 | 0.1023 | 0.1016 | 0.1019 |
| 5 | 0.1037 | 0.1034 | 0.1037 |
| 6 | 0.1050 | 0.1050 | 0.1053 |
| 7 | 0.1061 | 0.1065 | 0.1067 |
| 8 | 0.1072 | 0.1078 | 0.1080 |
| 9 | 0.1082 | 0.1090 | 0.1092 |
| 10 | 0.1092 | 0.1101 | 0.1103 |
| 11 | 0.1100 | 0.1111 | 0.1113 |
| 12 | 0.1109 | 0.1121 | 0.1122 |
使用rugarch包
- rugarch扩展包提供了更多的GARCH变种（改进）模型的估计功能， 我们用rugarch包估计intel股票月度对数收益率的GARCH模型，使用有偏t分布：
library(rugarch)## Loading required package: parallel##
## Attaching package: 'rugarch'## The following object is masked from 'package:stats':
##
## sigmaspec1 <- ugarchspec(
mean.model = list(
armaOrder=c(0,0),
include.mean=TRUE ),
variance.model = list(
model = "sGARCH", # standard GARCH model
garchOrder = c(1,1) ),
distribution.model="sstd" )
mod2ru <- ugarchfit(spec = spec1, data = ts.intel)
show(mod2ru)##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : sstd
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.013349 0.005344 2.4980 0.012491
## omega 0.001161 0.000559 2.0783 0.037684
## alpha1 0.104594 0.035883 2.9149 0.003558
## beta1 0.817923 0.056053 14.5919 0.000000
## skew 0.871842 0.062835 13.8750 0.000000
## shape 7.249884 2.107753 3.4396 0.000583
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.013349 0.005724 2.3321 0.019694
## omega 0.001161 0.000508 2.2865 0.022227
## alpha1 0.104594 0.030754 3.4010 0.000671
## beta1 0.817923 0.048372 16.9090 0.000000
## skew 0.871842 0.061014 14.2891 0.000000
## shape 7.249884 1.994299 3.6353 0.000278
##
## LogLikelihood : 328.1036
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.4509
## Bayes -1.3956
## Shibata -1.4513
## Hannan-Quinn -1.4291
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.5396 0.4626
## Lag[2*(p+q)+(p+q)-1][2] 0.7088 0.6032
## Lag[4*(p+q)+(p+q)-1][5] 1.8503 0.6542
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2528 0.6151
## Lag[2*(p+q)+(p+q)-1][5] 0.3020 0.9835
## Lag[4*(p+q)+(p+q)-1][9] 0.4598 0.9988
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.005876 0.500 2.000 0.9389
## ARCH Lag[5] 0.038484 1.440 1.667 0.9966
## ARCH Lag[7] 0.068420 2.315 1.543 0.9997
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.337
## Individual Statistics:
## mu 0.15846
## omega 0.13798
## alpha1 0.10628
## beta1 0.10705
## skew 0.04290
## shape 0.06739
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.17515 0.8610
## Negative Sign Bias 0.26118 0.7941
## Positive Sign Bias 0.07641 0.9391
## Joint Effect 0.09363 0.9926
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 16.27 0.6392
## 2 30 29.78 0.4249
## 3 40 35.28 0.6403
## 4 50 52.17 0.3517
##
##
## Elapsed time : 0.2774439结果与用fGARCH包计算的结果基本一致。
distribution.model的默认条件分布是"norm"， 即条件正态分布。rugarch也可以用plot()函数对建模拟合结果制作各种诊断图形， 如标准化残差相对于理论分布的QQ图：
plot(mod2ru, which=9)
4.6.6 两步估计法
例如，对Intel月对数收益率序列， 先用常数作为均值方程，计算残差\(a_t\):
hatmu <- mean(ts.intel); hatmu## [1] 0.0143273at <- ts.intel - hatmu
at2 <- at^2- 关于\(a_t^2\)用ARMA(1,1)估计：
mod4 <- arima(at2, order=c(1,0,1)); mod4##
## Call:
## arima(x = at2, order = c(1, 0, 1))
##
## Coefficients:
## ar1 ma1 intercept
## 0.9119 -0.7915 0.0161
## s.e. 0.0430 0.0635 0.0039
##
## sigma^2 estimated as 0.001223: log likelihood = 858.64, aic = -1709.28- 这样得到的误差波动率的估计为：
vola4 <- sqrt(at2 - mod4$residuals)- 与有偏t分布的GARCH拟合的波动率计算相关系数：
cor(vola3, vola4)## [1] 0.9976243- 两者十分相似。
4.6.7 IGARCH模型
GARCH模型可以写成(18.2)这样的关于\(a_t^2\)的ARMA形式， 其中\(\eta_t = a_t^2 - \sigma_t^2\)是模型的扰动， 是鞅差白噪声：
Intel股票的月对数收益率建立IGARCH(1,1)模型：
library(rugarch)
speci <- ugarchspec(
mean.model = list(
armaOrder=c(0,0),
include.mean=TRUE ),
variance.model = list(
model = "iGARCH", # itegrated GARCH model
garchOrder = c(1,1) ) )
mod5ru <- ugarchfit(spec = speci, data = xts.intel)
show(mod5ru)##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : iGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.009665 0.005340 1.8100 0.070298
## omega 0.000345 0.000182 1.8990 0.057561
## alpha1 0.127533 0.033626 3.7927 0.000149
## beta1 0.872467 NA NA NA
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.009665 0.006240 1.5488 0.121425
## omega 0.000345 0.000191 1.8060 0.070925
## alpha1 0.127533 0.031820 4.0079 0.000061
## beta1 0.872467 NA NA NA
##
## LogLikelihood : 308.1861
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.3747
## Bayes -1.3470
## Shibata -1.3748
## Hannan-Quinn -1.3638
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.6333 0.4261
## Lag[2*(p+q)+(p+q)-1][2] 0.8602 0.5459
## Lag[4*(p+q)+(p+q)-1][5] 1.8181 0.6620
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2326 0.6296
## Lag[2*(p+q)+(p+q)-1][5] 0.5702 0.9463
## Lag[4*(p+q)+(p+q)-1][9] 0.8666 0.9911
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.1295 0.500 2.000 0.7190
## ARCH Lag[5] 0.2649 1.440 1.667 0.9495
## ARCH Lag[7] 0.3838 2.315 1.543 0.9877
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.1698
## Individual Statistics:
## mu 0.22485
## omega 0.03416
## alpha1 0.21842
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.846 1.01 1.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.24275 0.8083
## Negative Sign Bias 0.06684 0.9467
## Positive Sign Bias 0.25664 0.7976
## Joint Effect 0.10895 0.9907
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 27.71 0.08914
## 2 30 37.08 0.14416
## 3 40 57.98 0.02571
## 4 50 59.38 0.14717
##
##
## Elapsed time : 0.07020092Intel股票月对数收益率的GARCH-M建模
- 用rugarch包估计GARCH-M模型：
library(rugarch)
specm <- ugarchspec(
mean.model = list(
armaOrder=c(0,0),
include.mean=TRUE,
archm=TRUE, archpow=2),
variance.model = list(
model = "sGARCH", # standard GARCH model
garchOrder = c(1,1) ) )
mod6ru <- ugarchfit(spec = specm, data = xts.intel)
show(mod6ru)##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000182 0.013692 -0.013286 0.989400
## archm 0.855502 0.949132 0.901352 0.367401
## omega 0.000944 0.000392 2.408992 0.015997
## alpha1 0.087242 0.026658 3.272661 0.001065
## beta1 0.849716 0.039235 21.657192 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.000182 0.015261 -0.01192 0.990489
## archm 0.855502 1.097457 0.77953 0.435667
## omega 0.000944 0.000465 2.02966 0.042391
## alpha1 0.087242 0.029544 2.95293 0.003148
## beta1 0.849716 0.041846 20.30575 0.000000
##
## LogLikelihood : 312.7599
##
## Information Criteria
## ------------------------------------
##
## Akaike -1.3863
## Bayes -1.3402
## Shibata -1.3866
## Hannan-Quinn -1.3681
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.6479 0.4209
## Lag[2*(p+q)+(p+q)-1][2] 0.8671 0.5434
## Lag[4*(p+q)+(p+q)-1][5] 2.1293 0.5882
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1373 0.7110
## Lag[2*(p+q)+(p+q)-1][5] 0.1496 0.9958
## Lag[4*(p+q)+(p+q)-1][9] 0.3174 0.9997
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.0002646 0.500 2.000 0.9870
## ARCH Lag[5] 0.0167632 1.440 1.667 0.9990
## ARCH Lag[7] 0.0530468 2.315 1.543 0.9999
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.5296
## Individual Statistics:
## mu 0.15303
## archm 0.17083
## omega 0.07591
## alpha1 0.09511
## beta1 0.07486
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.2498 0.8029
## Negative Sign Bias 0.4723 0.6370
## Positive Sign Bias 0.1415 0.8875
## Joint Effect 0.2692 0.9657
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 17.35 0.5661
## 2 30 22.76 0.7876
## 3 40 30.77 0.8236
## 4 50 49.47 0.4544
##
##
## Elapsed time : 0.3301179标普500指数月超额收益率的GARCH-M建模
- 考虑标普500指数从1926年到1991年的月超额收益率。 共792个观测。
x.sp <- scan("D:/齐安静 教学/时间序列分析/北大/ftsdata/sp500.txt", quiet=TRUE)*100
ts.sp <- ts(x.sp, start=c(1926,1), frequency = 12)
plot(ts.sp, main="标普500的月超额收益率(%)", xlab="年", ylab="超额收益率")
- 用GARCH(1,1)模型：
library(rugarch)
spec.sp1 <- ugarchspec(
mean.model = list(
armaOrder=c(0,0),
include.mean=TRUE ),
variance.model = list(
model = "sGARCH", # standard GARCH model
garchOrder = c(1,1) ) )
mod7ru <- ugarchfit(spec = spec.sp1, data = ts.sp)
show(mod7ru)##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.74497 0.153760 4.8450 0.000001
## omega 0.80400 0.284209 2.8289 0.004671
## alpha1 0.12226 0.022102 5.5315 0.000000
## beta1 0.85434 0.021813 39.1664 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.74497 0.171721 4.3382 0.000014
## omega 0.80400 0.336109 2.3921 0.016753
## alpha1 0.12226 0.028162 4.3413 0.000014
## beta1 0.85434 0.026423 32.3329 0.000000
##
## LogLikelihood : -2377.839
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.0147
## Bayes 6.0384
## Shibata 6.0147
## Hannan-Quinn 6.0238
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.6270 0.4284
## Lag[2*(p+q)+(p+q)-1][2] 0.6322 0.6347
## Lag[4*(p+q)+(p+q)-1][5] 2.7320 0.4582
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.622 0.4303
## Lag[2*(p+q)+(p+q)-1][5] 1.867 0.6502
## Lag[4*(p+q)+(p+q)-1][9] 3.671 0.6450
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.6535 0.500 2.000 0.4189
## ARCH Lag[5] 1.3390 1.440 1.667 0.6356
## ARCH Lag[7] 1.9549 2.315 1.543 0.7269
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.0291
## Individual Statistics:
## mu 0.08926
## omega 0.20235
## alpha1 0.27140
## beta1 0.16486
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 3.1027 0.0019860 ***
## Negative Sign Bias 0.9191 0.3583444
## Positive Sign Bias 0.7084 0.4788761
## Joint Effect 17.6088 0.0005296 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 30.73 0.04324
## 2 30 40.27 0.07958
## 3 40 48.40 0.14373
## 4 50 61.41 0.10989
##
##
## Elapsed time : 0.1409812模型很接近IGARCH条件。
- 尝试对标普500超额收益率拟合GARCH-M(1,1)模型：
library(rugarch)
spec.sp2 <- ugarchspec(
mean.model = list(
armaOrder=c(0,0),
include.mean=TRUE,
archm=TRUE, archpow=2),
variance.model = list(
model = "sGARCH", # standard GARCH model
garchOrder = c(1,1) ) )
mod8ru <- ugarchfit(spec = spec.sp2, data = ts.sp)
show(mod8ru)##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.542035 0.236018 2.2966 0.021643
## archm 0.010081 0.008885 1.1347 0.256520
## omega 0.829651 0.293213 2.8295 0.004662
## alpha1 0.123124 0.022286 5.5247 0.000000
## beta1 0.852261 0.022400 38.0478 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.542035 0.243968 2.2217 0.026300
## archm 0.010081 0.008913 1.1311 0.258009
## omega 0.829651 0.347876 2.3849 0.017083
## alpha1 0.123124 0.027802 4.4286 0.000009
## beta1 0.852261 0.027061 31.4946 0.000000
##
## LogLikelihood : -2377.192
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.0156
## Bayes 6.0451
## Shibata 6.0156
## Hannan-Quinn 6.0270
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.7246 0.3946
## Lag[2*(p+q)+(p+q)-1][2] 0.7501 0.5869
## Lag[4*(p+q)+(p+q)-1][5] 3.0243 0.4026
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.7589 0.3837
## Lag[2*(p+q)+(p+q)-1][5] 1.8690 0.6497
## Lag[4*(p+q)+(p+q)-1][9] 3.4842 0.6771
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.5048 0.500 2.000 0.4774
## ARCH Lag[5] 1.0892 1.440 1.667 0.7066
## ARCH Lag[7] 1.6862 2.315 1.543 0.7833
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.327
## Individual Statistics:
## mu 0.06463
## archm 0.03109
## omega 0.21419
## alpha1 0.26216
## beta1 0.16482
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 3.5833 3.601e-04 ***
## Negative Sign Bias 1.1596 2.466e-01
## Positive Sign Bias 0.6211 5.347e-01
## Joint Effect 22.1806 5.982e-05 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 23.86 0.20163
## 2 30 41.11 0.06741
## 3 40 40.63 0.39861
## 4 50 56.86 0.20554
##
##
## Elapsed time : 0.3657961