How can I use the results of GARCH in order to improve a forecast? I am kind of confused with what I should actually do with predicted volatility values that I obtained via a ARCH/GARCH model other than feeling happy that I know when volatility rises/falls. Is there a way that I can incorporate the predicted volatility values via a GARCH/ARCH model into a prediction model for my actual time series or is what I am saying erroneous? I use R as my primary tool.
 A: Garch models can help you construct density forecats; the model will show you the expected outcome while the garch component gives you the uncertainty/volatiiity around the outcome, see e.g. http://www2.warwick.ac.uk/fac/soc/economics/staff/academic/wallis/publications/taywallis_jof_00.pdf
First to make things simple, assume I know, the true parameters in the model:
$y_t = \alpha + \beta y_{t-1} +  \sigma_t\varepsilon_t$
The best prediction in terms of root mean squared error is:
$\hat{y}_{t+1|t} = \alpha + \beta y_t$
My prediction error is:
$\hat{y}_{t+1|t}-y_{t+1} = \sigma_{t+1}\varepsilon_{t+1}$ so that $E(\hat{y}_{t+1|t}-y_{t+1})^2 = E(\sigma_{t+1}^2)$, if $V(\varepsilon_{t+1})=1$
The Garch model can help me predict this value, for instance with an ARCH(1) model I can use:
$\hat{\sigma}_{t+1|t}^2 = \alpha_0 + \alpha_1 \varepsilon_t^2$
Since, I don't know $\alpha,\beta,\alpha_0,\alpha_1$ I have to substitute them with their estimated counterparts $\hat{\alpha},\dots$ and the fitted $\hat{\varepsilon}_t$
This will add uncertainty to my forecast: I don't know the innovation $\varepsilon_{t+1}$ and I don't know the parameters. The two are independent (assuming the innovations are iid) so since the model is non-linear you can sample the parameters from a normal distribution centered around the estimates with variance matrix of the estimator as a covariance matrix, you also compute the fitted $\hat{\varepsilon}_t$ for each draw and draw $\varepsilon_{t+1}$ from a standard normal distribution. Using this you can compute a possible realization of $y_{t+1}$, you can do this many many times (Monte Carlo). Then you can compute the 2.5 and 97.5% quantiles of these draws and this gives a 95% confidence interval for your prediction.
I hope it helps!!!
