GARCH模型

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GARCH模型的定义

AR-GARCH模型

w<-read.table("D:/R-TT/book4/4R/data/file23.csv",sep=",",header = T) x<-ts(w$exchange_rates,start=c(1979,12,31),frequency = 365) plot(x) 
 
 
  
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外币对美元日兑换率序列时序图 
对差分序列性质的考察

plot(diff(x)) 
 
 
  
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外币对美元日兑换率序列1阶差分时序图 
外币对美元日兑换率序列1阶差分自相关图

acf(diff(x)) 
 
 
  
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外币对美元日兑换率序列1阶差分自相关图

pacf(diff(x)) 
 
 
  
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外币对美元日兑换率序列1阶差分偏自相关图 
序列时序图显示序列非平稳，有明显的趋势特征，差分后序列时序图显示趋势消除，但是有明显的集群效应，所以分析该序列需要同时提取水平相关信息与波动相关信息。 
水平信息的提取是考察差分后的自相关性与偏相关性，拟合ARIMA(0,1,1)。

• 水平相关信息提取

#水平相关信息提取，拟合ARIMA(0,1,1)模型 x.fit<-arima(x,order = c(0,1,1)) x.fit Call: arima(x = x, order = c(0, 1, 1)) Coefficients: ma1 0.0357 s.e. 0.0143 sigma^2 estimated as 0.0002007: log likelihood = 13545.61, aic = -27087.22 
 
 
  
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> #残差白噪声检验 > for (i in 1:6) print(Box.test(x.fit$residual,type = "Ljung-Box",lag=i)) Box-Ljung test data: x.fit$residual X-squared = 0.0005354, df = 1, p-value = 0.9815 Box-Ljung test data: x.fit$residual X-squared = 0.55102, df = 2, p-value = 0.7592 Box-Ljung test data: x.fit$residual X-squared = 2.6528, df = 3, p-value = 0.4483 Box-Ljung test data: x.fit$residual X-squared = 3.3062, df = 4, p-value = 0.5079 Box-Ljung test data: x.fit$residual X-squared = 6.8276, df = 5, p-value = 0.2338 Box-Ljung test data: x.fit$residual X-squared = 6.8306, df = 6, p-value = 0.3368  
 
 
  
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该拟合模型的残差白噪声检验显示该模型显著成立，利用该拟合模型可以预测列未来的水平。

#水平预测，并绘制预测图 library(forecast) x.fore<-forecast(x.fit,h=365) plot(x.fore) 
 
 
  
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外币对美元日兑换率序列日预测图

• 波动相关信息提取 
  波动信息的提取首先是考察ARIMA(0,1,1)模型的残差平方序列的异方差特征。
  

#条件异方差检验(Portmanteau Q检验) for (i in 1:6) print(Box.test(x.fit$residual^2,type = "Ljung-Box",lag=i)) Box-Ljung test data: x.fit$residual^2 X-squared = 82.803, df = 1, p-value < 2.2e-16 Box-Ljung test data: x.fit$residual^2 X-squared = 237.9, df = 2, p-value < 2.2e-16 Box-Ljung test data: x.fit$residual^2 X-squared = 343.33, df = 3, p-value < 2.2e-16 Box-Ljung test data: x.fit$residual^2 X-squared = 490.84, df = 4, p-value < 2.2e-16 Box-Ljung test data: x.fit$residual^2 X-squared = 602.1, df = 5, p-value < 2.2e-16 Box-Ljung test data: x.fit$residual^2 X-squared = 841.96, df = 6, p-value < 2.2e-16 
 
 
  
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波动信息的提取首先是考察ARIMA(0，1，1)模型的残差平方序列的异方差特征，Portmanteau Q检验显示残差序列显著方差非齐性，且具有长期相关性，所以构造GARCH（1，1）模型，并根据该模型的拟合结果绘制波动的95%置信区间。

#拟合GARCH(1,1)模型 r.fit<-garch(x.fit$residual,order=c(1,1)) summary(r.fit) Call: garch(x = x.fit$residual, order = c(1, 1)) Model: GARCH(1,1) Residuals: Min 1Q Median 3Q Max -4.83074 -0.58407 0.02616 0.58758 4.54060 Coefficient(s): Estimate Std. Error t value Pr(>|t|) a0 2.133e-06 3.014e-07 7.077 1.48e-12 * a1 7.623e-02 5.456e-03 13.972 < 2e-16 * b1 9.144e-01 6.015e-03 152.009 < 2e-16 * --- Signif. codes: 0 ‘*’ 0.001 ‘’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Diagnostic Tests: Jarque Bera Test data: Residuals X-squared = 319.23, df = 2, p-value < 2.2e-16 Box-Ljung test data: Squared.Residuals X-squared = 0.28019, df = 1, p-value = 0.5966` 
 
 
  
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#绘制波动置信区间 r.pred<-predict(r.fit) plot(r.pred) 
 
 
  
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外币对美元日兑换率序列残差波动置信区间

GARCH衍生模型