EDIT :
I read more about it and I get some help with someone else, here is the correct answer :
The density forecast is the predictive likelihood value of the process
estimated at the realized value computed in a one step ahead way.
Thus for instance for a standard arma garch process with normal errors:
- You forecast the mean $u^{f}_{t|t-1}$ and variance $v^{f}_{t|t-1}$ process at time t-1 for time t
- the predictive density forecast for time t of the realized value $u_{t}$ is $N(u^{f}_{t|t-1},v^{f}_{t|t-1})$
- the predictive density forecast $u_{t}\sim N(u^{f}_{t|t-1},v^{f}_{t|t-1}) $ is equivalent to the predictive residuals density forecast : $r_{t} = u_{t}-u^{f}_{t|t-1}\sim N(0,v^{f}_{t|t-1}) $
- the density forecast is the density of $r_{t}$ with respect to a $N(0,v^{f}_{t|t-1}) $
Note that it is very similar to the "usual" likelihood except you are estimating the model in a one step ahead way (the parameters are re-estimated at every step)
Previous Post :
It is not a "basic question", If I am correct :
First you estimate your model on the return series and obtains parameters. You must estimate your model in such a way you obtain one-step ahead errors (that I will call computed errors in what follow) and associated time-serie of the predictive errors distributions parameters: $\hat{\mu_{t}}$ and $ \hat{\sigma_{t}^{2}}$ (note : these are not the parameters of the mean and variance processes but parameters of the error distribution).
I would take the original return serie minus the fitted returns to obtain the computed errors :
$$\hat{e_{t}} = r_{t} - \hat{r_{t}}$$
These computed errors should behave accordingly your predictive (errors) density which is defined by parameters you obtain in the first step ($\hat{\mu_{t}}, \hat{\sigma_{t}}$) : Note the subscript $_{t}$ for parameters !
Then I would compute the predictive density of these residuals using parameters obtained in the first step estimation, indeed if your forecasts are accurate the
density of $\hat{e_{t}}$ should ideally be equal to the predictive density defined as: normal($\hat{\mu_{t}}, \hat{\sigma_{t}}$).
If it fit perfectly (they have the same mean, variance) then the density forecast will returns a high mass . If the model is misspecified, the errors $\hat{e_{t}}$ will fall outside of the range implied by your predictive error density and then it will assign very small probability.
So in your case I will use the following function (again note the time subscript)
Density forecast ($\hat{e_{t}}$) = normpdf($\hat{e_{t}}, \hat{\mu_{t}}, \hat{\sigma_{t}}$)