最近我被要求撰写关于金融时间序列的copulas的调查。 从读取数据中获得各种模型的描述,包括一些图形和统计输出。
-
> oil = read.xlsx(temp,sheetName =“DATA”,dec =“,”)
然后我们可以绘制这三个时间序列
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1 1997-01-10 2.73672 2.25465 3.3673 1.5400 -
2 1997-01-17 -3.40326 -6.01433 -3.8249 -4.1076 -
3 1997-01-24 -4.09531 -1.43076 -6.6375 -4.6166 -
4 1997-01-31 -0.65789 0.34873 0.7326 -1.5122 -
5 1997-02-07 -3.14293 -1.97765 -0.7326 -1.8798 -
6 1997-02-14 -5.60321 -7.84534 -7.6372 -11.0549
这个想法是在这里使用一些多变量ARMA-GARCH过程。这里的启发式是第一部分用于模拟时间序列平均值的动态,第二部分用于模拟时间序列方差的动态。
本文考虑了两种模型
- 关于ARMA模型残差的多变量GARCH过程(或方差矩阵动力学模型)
- 关于ARMA-GARCH过程残差的多变量模型(基于copula)
因此,这里将考虑不同的序列,作为不同模型的残差获得。我们还可以将这些残差标准化。
ARMA模型
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> fit1 = arima(x = dat [,1],order = c(2,0,1)) -
> fit2 = arima(x = dat [,2],order = c(1,0,1)) -
> fit3 = arima(x = dat [,3],order = c(1,0,1)) -
> m < - apply(dat_arma,2,mean) -
> v < - apply(dat_arma,2,var) -
> dat_arma_std < - t((t(dat_arma)-m)/ sqrt(v))
ARMA-GARCH模型
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> fit1 = garchFit(formula = ~arma(2,1)+ garch(1,1),data = dat [,1],cond.dist =“std”) -
> fit2 = garchFit(formula = ~arma(1,1)+ garch(1,1),data = dat [,2],cond.dist =“std”) -
> fit3 = garchFit(formula = ~arma(1,1)+ garch(1,1),data = dat [,3],cond.dist =“std”) -
> m_res < - apply(dat_res,2,mean) -
> v_res < - apply(dat_res,2,var) -
> dat_res_std = cbind((dat_res [,1] -m_res [1])/ sqrt(v_res [1]),(dat_res [,2] -m_res [2])/ sqrt(v_res [2]),(dat_res [ ,3] -m_res [3])/ SQRT(v_res [3]))
多变量GARCH模型
可以考虑的第一个模型是协方差矩阵的多变量EWMA,
波动性
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> emwa_series_vol = function(i = 1){ -
+ lines(Time,dat_arma [,i] + 40,col =“gray”) -
+ j = 1 -
+ if(i == 2)j = 5 -
+ if(i == 3)j = 9
隐含相关性
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> emwa_series_cor = function(i = 1,j = 2){ -
+ if((min(i,j)== 1)&(max(i,j)== 2)){ -
+ a = 1; B = 9; AB = 3} -
+ r = ewma $ Sigma.t [,ab] / sqrt(ewma $ Sigma.t [,a] * -
+ ewma $ Sigma.t [,b]) -
+ plot(Time,r,type =“l”,ylim = c(0,1)) -
+}
多变量GARCH,即BEKK(1,1)模型,例如使用:
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> bekk = BEKK11(dat_arma) -
> bekk_series_vol function(i = 1){ -
+ plot(Time, $ Sigma.t [,1],type =“l”, -
+ ylab = (dat)[i],col =“white”,ylim = c(0,80)) -
+ lines(Time,dat_arma [,i] + 40,col =“gray”) -
+ j = 1 -
+ if(i == 2)j = 5 -
+ if(i == 3)j = 9 -
> bekk_series_cor = function(i = 1,j = 2){ -
+ a = 1; B = 5; AB = 2} -
+ a = 1; B = 9; AB = 3} -
+ a = 5; B = 9; AB = 6} -
+ r = bk $ Sigma.t [,ab] / sqrt(bk $ Sigma.t [,a] * -
+ bk $ Sigma.t [,b])
从单变量GARCH模型中模拟残差
第一步可能是考虑残差的一些静态(联合)分布。单变量边缘分布是
边缘密度的轮廓(使用双变量核估计器获得)
也可以将copula密度可视化(上面有一些非参数估计,下面是参数copula)
-
> copula_NP = function(i = 1,j = 2){ -
+ n = nrow(uv) -
+ s = 0.3 -
+ norm.cop < - normalCopula(0.5) -
+ norm.cop < - normalCopula(fitCopula(norm.cop,uv)@estimate) -
+ dc = function(x,y)dCopula(cbind(x,y),norm.cop) -
+ ylab = names(dat)[j],zlab =“copule Gaussienne”,ticktype =“detailed”,zlim = zl) -
+ -
+ t.cop < - tCopula(0.5,df = 3) -
+ t.cop < - tCopula(t.fit [1],df = t.fit [2]) -
+ ylab = names(dat)[j],zlab =“copule de Student”,ticktype =“detailed”,zlim = zl) -
+}
可以考虑这个
函数,
计算三个序列的的经验版本,并将其与一些参数版本进行比较,
-
> -
> lambda = function(C){ -
+ l = function(u)pcopula(C,cbind(u,u))/ u -
+ v = Vectorize(l)(u) -
+ return(c(v,rev(v))) -
+} -
> -
> graph_lambda = function(i,j){ -
+ X = dat_res -
+ U = rank(X [,i])/(nrow(X)+1) -
+ V = rank(X [,j])/(nrow(X)+1) -
+ normal.cop < - normalCopula(.5,dim = 2) -
+ t.cop < - tCopula(.5,dim = 2,df = 3) -
+ fit1 = fitCopula(normal.cop,cbind(U,V),method =“ml”) -
d(U,V),method =“ml”) -
+ C1 = normalCopula(fit1 @ copula @ parameters,dim = 2) -
+ C2 = tCopula(fit2 @ copula @ parameters [1],dim = 2,df = trunc(fit2 @ copula @ parameters [2])) -
+
但人们可能想知道相关性是否随时间稳定。
-
>function(i = 1,j = 2, -
+ nom_arg =“Pearson”){ -
+ uv = dat_arma [,c(i,j)] -
nom_arg))[1,2] -
+} -
> time_varying_correl_2(1,2) -
> time_varying_correl_2(1,2,“spearman”) -
> time_varying_correl_2(1,2,“kendall”)
斯皮尔曼与时变排名相关系数
或肯德尔 相关系数
为了模型的相关性,考虑DCC模型(S)
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> m2 = dccFit(dat_res_std) -
>type =“Engle”) -
> R2 = m2 $ rho.t -
> R3 = m3 $ rho.t
要获得一些预测, 使用例如
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> garch11.spec = ugarchspec(mean.model = list(armaOrder = c(2,1)),variance.model = list(garchOrder = c(1,1),model =“GARCH”)) -
> dcc.garch11.spec = dccspec(uspec = multispec(replicate(3,garch11.spec)),dccOrder = c(1,1), -
distribution =“mvnorm”) -
> dcc.fit = dccfit(dcc.garch11.spec,data = dat) -
> fcst = dccforecast(dcc.fit,n.ahead = 200)