Forecasting returns using GARCH model alone I need to know if I gave a GARCH model the returns of prices, how could I get the forecasts for the returns not the volatility in case of using GARCH without ARIMA (because in the equation of GARCH model there is an error term which is random)? But in some papers the GARCH is used for forecasting the return not the sigma. How can I do it in R, say, "fGarch" package? 
 A: A model for the returns $r_t$ with a GARCH structure for the conditional variance will look like this:
\begin{aligned} 
r_t &= \mu_t + u_t, \\
u_t &= \sigma_t \varepsilon_t, \\
\sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \\
\varepsilon_t &\sim i.i.d.(0,1),
\end{aligned}
where $\mu_t$ is the conditional mean of $r_t$ which could be e.g. a constant or an ARMA process. (Here we have a GARCH(1,1) model, but the extension to GARCH(p,q) is trivial.)
An optimal (under square loss) point forecast of $r_{t+h}$ for some $h>0$, given the available information $I_t$, is
\begin{aligned}
\mathbb{E}(r_{t+h}|I_t) &= \mathbb{E}(\mu_{t+h}+u_{t+h}|I_t) \\
&= \mathbb{E}(\mu_{t+h}|I_t) + \mathbb{E}(u_{t+h}|I_t) \\
&= \mathbb{E}(\mu_{t+h}|I_t) + 0 \\
&= \mathbb{E}(\mu_{t+h}|I_t). \\
\end{aligned}
Since we normally do not know $\mathbb{E}(\mu_{t+h}|I_t)$, we take a forecasted value from the model, $\hat\mu_{t+h|t}$. So in practice the point forecast is $\hat\mu_{t+h|t}$.
In case $\mu_t=\mu$ is a constant (could be zero, could be nonzero), the optimal (under square loss) point forecast of $r_{t+h}$ for some $h>0$ is $\mathbb{E}(\mu_{t+h}|I_t)=\mu$. In practice you would take the fitted intercept $\hat\mu$ from the conditional mean model as the point forecast. 
Example with R package "fGarch":
library(fGarch)
model = garchFit(formula = ~ garch(1, 1), data = dem2gbp, cond.dist = "norm", include.mean = TRUE)
fcst=predict(model,n.ahead=5)
mean.fcst=fcst$meanForecast

The last line saves the point forecasts in an object called mean.fcst.
Check coef(model) to verify that the forecasts equal the fitted intercept mu.
