# Can the GARCH model predict other things than only volatility?

I was reading that ARIMA models are used to model the mean whereas GARCH models are usually used to model the conditional variance (i.e. volatility). Is it possible that the GARCH model can somehow be modified to predict the future value of the time series itself instead of the future volatility?

E.g. if you have historical data on the total amount of rain falls every week - could a GARCH model be used to predict how much rain will fall next week? Or is this impossible?

Thanks!

A typical GARCH(r,s) model looks something like this: \begin{aligned} x_t &= \mu_t + u_t, \\ \mu_t &= \dots, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &\sim i.i.D(0,1), \end{aligned} where $$D$$ is some probability distribution with zero mean and unit variance.
GARCH models the entire conditional distribution of the dependent variable $$x_t$$, conditional on the history of $$x_t$$. Out of that you can derive a point forecast based on the loss function you are facing. E.g. if your loss is quadratic (a.k.a. square loss), the optimal point forecast is the conditional mean. Given an estimated GARCH model, the model's conditional mean equation $$\mu_t =\dots$$ will be sufficient for that.