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
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
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
> 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
> 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 \[3\]))
可以考虑的第一个模型是协方差矩阵的多变量EWMA,
> ewma = EWMAvol(dat\_res\_std,lambda = 0.96)
波动性
> 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
> 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:
> 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 \[,b\])
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。
> time\_varying\_correl_2 = 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)
> m2 = dccFit(dat\_res\_std)
> m3 = 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
> 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)
本文摘选《R语言ARMA-GARCH-COPULA模型和金融时间序列案例》