This paper presented at UseR! 2009 seems to address a similar problem
http://www.r-project.org/conferences/useR-2009/slides/Roustant+Ginsbourger+Deville.pdf
It suggests the DiceKriging package http://cran.r-project.org/web/packages/DiceKriging/index.html
In particular, check the functions km and predict.
Here is a an example of three dimensional interpolation. It looks to be straightforward to generalise.
x <- c(0, 0.4, 0.6, 0.8, 1)
y <- c(0, 0.2, 0.3, 0.4, 0.5)
z <- c(0, 0.3, 0.4, 0.6, 0.8)
model <- function(param){
2*param[1] + 3*param[2] +4*param[3]
}
model.in <- expand.grid(x,y,z)
names(model.in) <- c('x','y','z')
model.out <- apply(model.in, 1, model)
# fit a kriging model
m.1 <- km(design=model.in, response=model.out, covtype="matern5_2")
# estimate a response
interp <- predict(m.1, newdata=data.frame(x=0.5, y=0.5, z=0.5), type="UK", se.compute=FALSE)
# check against model output
interp$mean
# [1] 4.498902
model(c(0.5,0.5,0.5))
# [1] 4.5
# check we get back what we put in
interp <- predict(m.1, newdata=model.in, type="UK", se.compute=FALSE)
all.equal(model.out, interp$mean)
# TRUE
ARCH postestimation help file explains it all. You will most likely need
predict hat_volatility, variance
or
predict hat_volatility_factor, het
depending on what exactly you mean by ``volatility''. The former is the full prediction, the latter is the multiplier that goes in front of the $\hat\sigma^2$.
Best Answer
I don't now about Matlab but here is what I can tell about modelling the conditional variance-covariance matrix of a multivariate time series.
There are two nice overview papers of multivariate GARCH models:
I have personally tried the DCC model which is simple and easy to estimate (the latter property becomes important when you are working with more than two or three variables). The DCC model is also quite flexible with respect to the model choice for the variances of each of the variables (the diagonal elements of the variance-covariance matrix). You can, for example, choose an EGARCH model for one of the series, a FIGARCH for another etc., ect. There are also extensions of the DCC model allowing for asymmetries (the ADCC model)
or different dynamics in different groups of time series (the FDCC model)
The latter will be especially relevant if you are analyzing groups of assets where the assets within a group are somehow similar. However, DCC has received some serious critique:
Nevertheless, it is a widely used model with the original paper
having received over 3500 citations (as counted in Google Scholar).
The R implementation in "rmgarch" package is very good, in my opinion: