Evaluating GARCH Density Forecasts at Realized Returns

# Density forecast of a GARCH model


# Density forecast of a GARCH model












I am currently working on developing a series of density forecasts and I am encountering some problems. I am working on weekly S&P 500 returns and the returns process is described as

$r_{t} = \mu + \delta r_{t-1} + h_{t}z_{t}$ where $z_{t}$ comes from the Gaussian distribution.

I am forecasting the returns and volatility of the series using the ARMAX-GARCH-K Toolbox in Matlab. Initially I estimate the ARMA(1,0)-GARCH(1,1) model and obtain the one-step ahead forecast of the returns and volatility. I obtain the $\mu$ parameter from the parameters of the GARCH model.

As far as I understand the next step to obtain the density forecast (assuming Gaussian distribution) is to use the pdf of Gaussian distribution, so the

> normpdf(x, mu, sigma)

function in Matlab.

And here is the essence of my trouble. To obtain the density forecast should I use the actual observed return as x or the point forecast from the garch model? And should I use the $\mu$ parameter from the GARCH model as input for the mu parameter of normpdf and the forecasted volatility as as sigma in the normpdf function?

I now this may be a basic question but I cannot find any elementary examples on the internet.

Kind regards

## Answer by Malick (score 4, accepted)

https://quant.stackexchange.com/a/21911

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($\hat{e_{t}}, \hat{\mu_{t}}, \hat{\sigma_{t}}$)