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I am having some troubles obtaining density forecasts of any returns series. As I couldn't find any numerical examples on the Internet, I would like to ask you guys for some help.

My goal is to obtain density forecasts using a GARCH model (in R). Using the basic GARCH model (from the rugarch package) I can obtain a series of return forecasts or volatility forecasts (and assume a constant mean).What I do not understand however is how to obtain the density forecasts from the information I already have. Having the return and volatility forecasts what is the next step to obtain the density forecasts? I am not asking for a piece of code to calculate it per se but rather for explaining what are the steps necessary to obtain the density forecast (predicted density of a GARCH model).

I know this may be a simple question but any help would be really appreciated.

Kind regards

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Lets start with the set up of a simple GARCH(1,1) model $$ y_t=\sigma_t\varepsilon_t,\;\;\varepsilon_t \stackrel{iid}{\sim}N(0,1) $$ $$ Var(y_t | y_{t-1},\sigma_{t-1}) = \sigma^2_t = \alpha_0 + \alpha_1 y^2_{t-1}+ \beta_1\sigma^2_{t-1} $$ So we know $$ y_t|y_{t-1},\sigma_{t-1} \sim N(0,Var(y_t | y_{t-1})) \quad(1) $$

$(1)$ gives you the 1-step ahead predictive/forecast density and the best predictive density in the GARCH(1,1). Hopefully that makes sense.

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