# ARMA-GARCH-COPULA model and financial time series case | with code data

Recently I was asked to write a survey on copulas for financial time series

A description of the various models is obtained from the read data, including some graphical and statistical output.

`> oil = read.xlsx(temp，sheetName ="DATA"，dec ="，")`
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We can then plot the three time series

```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```
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The idea is to use some multivariate ARMA-GARCH process here. The heuristic here is that the first part is used to model the dynamics of the time series mean and the second part is used to model the dynamics of the time series variance.

This paper considers two models

• Multivariate GARCH process on ARMA model residuals (or variance matrix dynamics model)
• Multivariate model (copula-based) on ARMA-GARCH process residuals

Therefore, different series will be considered here, obtained as residuals of different models. We can also normalize these residuals.

ARMA model

```> 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))```
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ARMA-GARCH model

```> 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 )/ sqrt(v_res )，(dat_res [，2] -m_res )/ sqrt(v_res )，(dat_res [ ，3] -m_res )/ SQRT(v_res ))```
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# Multivariate GARCH model

The first model that can be considered is the multivariate EWMA of the covariance matrix,

`> ewma = EWMAvol(dat_res_std，lambda = 0.96)`
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volatility

```> 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```
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implicit correlation

```> 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))
+}```
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A multivariate GARCH, i.e. BEKK(1,1) model, for example using:

```> 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])```
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# Simulate residuals from univariate GARCH model

A first step might be to consider some static (joint) distribution of the residuals. The univariate marginal distribution is

Profile of edge density (obtained using bivariate kernel estimator)

It is also possible to visualize the copula density (some nonparametric estimates above, parametric copula below)

```> 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 ，df = t.fit )

+ ylab = names(dat)[j]，zlab ="copule de Student"，ticktype ="detailed"，zlim = zl)
+}```
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can consider this

function,

Compute empirical versions of the three sequences and compare them to some parametric versions,

```>

> 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 ，dim = 2，df = trunc(fit2 @ copula @ parameters ))
+```
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But one might wonder whether the correlation is stable over time.

```> 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")```
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Spearman and time-varying rank correlation coefficient

or Kendall correlation coefficient

For model relevance, consider the DCC model(S)

```> m2 = dccFit(dat_res_std)
> m3 = dccFit(dat_res_std，type ="Engle")
> R2 = m2 \$ rho.t
> R3 = m3 \$ rho.t```
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To get some predictions, use e.g.

```> 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)```
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