R: Density Forecasts using rugarch

Question

I would like to construct a density forecast using a GARCH model. Is it possible to use rugarch in R to construct these? For example, using a ARMA(1,1)-GARCH(1,1) normal distribution model. I'll provide a simple example to start using data in rugarch (sp500ret). Is there anyway to proceed using this package or do I need to implement by own code? I know there are a number of methods using simulation of the GARCH process to construct density forecasts such as the method to construct historical densities presented in Taylor (2005) Asset Price Dynamics, Volatility, and Prediction. But I wonder if there's a way I can use rugarch to do this for me. Thank you in advance.

library(rugarch)
garchspec <- ugarchspec(mean.model = list(armaOrder = c(1,1)),
variance.model = list(model = "sGARCH"),
distribution.model = "norm")
garchfit <- ugarchfit(data = sp500ret, spec = garchspec)

Answer 1

Your GARCH model will produce a density forecast as is. It will be the normal density with mean coming from the ARMA equation and variance coming from the GARCH equation. This holds both one and multiple steps ahead, since the the dependent variable is a sum of i.i.d. normal random variables (the standardized innovations) shifted by some means (ARMA) and scaled by some variances (GARCH) which is still a normal random variable. You will find more detail in

• "How to make $h$-step interval forecasts from an ARMA-GARCH model?" and
• "What is the difference between GARCH and ARMA?".

If the algebraic details are cumbersome, you could always simulate with ugarchsim or ugarchpath and take the empirical density of the simulated series as your density forecast.