GARCH fitted parametric distributions for copula fitting

z = residuals(fit, standardize=TRUE)

pdist("std", z, 0, 1, shape = coef(fit)["shape"])

pit(fit)

Hello,

I would greatly appreciate any insights into the problem described below,
regarding using the data obtained from applying the functions of the
'rugarch' package into those from the 'copula' package.

I am endeavouring an investigation of dependencies between variables (i.e.
stock quotes, exchange rates etc.) using copula functions. I first model
each of my variables as an ARMA-GARCH process (or EGARCH/GJR to account for
asymmetries) and then use these models in order to determine the
best-fitting copula.

Take for example the case of stock quotes. After transforming them to log
returns ( using diff(log(stockData)) ), and investigating their correlation
with Kendall's thau and Spearman's rho, I fit an ARMA-GARCH model to each
variable (resulting in the so-called margin functions):

## Choose conditional mean model for each variable i in the retStock xts
object
autoarfima(data = retStock[,i], ar.max = 3, ma.max = 3, criterion = "AIC",
method = "full", distr = "std")

## Choose conditional variance model by comparing AICs for all combinations
of GARCH orders limited to 3, with the ARMA(p,q) mean model selected above
for (i in 1:3){
      for (j in 1:3){

spec <- ugarchspec(variance.model=list(model="sGARCH", garchOrder =
c(i,j)), mean.model=list(armaOrder=c(p,q), distribution.model = "std")

fit <- ugarchfit(spec = spec, data = retStock[,n])

# record AIC for overall comparison after exiting the loop
}

}

Keeping the uGARCHfit object with the best-fitting model, I move on to the
bivariate copulas. For completing the "Inference Functions for Margins"
method, I specifically need to input the observations from the *fitted
parametric marginal distribution functions*, for which I use the
*@fit$fitted.values* of each uGARCHfit object.

However, the fitted values of the two models are, for some variables,
negatively correlated (as indicated by Kendall's tau and Spearman's rho),
whereas the original observations were positively correlated, leading to
the copula fitting breaking down (for Gumbel and Clayton copulas, but I
won't go into copula details, as my question only concerns the output data
from the 'rugarch' package).

My overall question is, considering my approach and the need for inputting
fitted parametric marginal distribution functions, are the fitted values
from the uGARCHfit objects the right input data to be taking further into
the copula functions? Why are the fitted values negatively correlated, when
the originals were positively correlated?

I am running everything in RStudio 0.98.501 with R 3.0.2 on Windows 8.1. If
any further information is needed regarding my system specs or other
implementation details, let me know and I'll provide them as soon as
possible.

Thank you very much for your assistance,
Sebastian

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