Forecasting levels through GARCH model

Question

I am working on a model which describes returns only as a constant plus a garch structure(in particular a garch-midas, but this is not crucial for my question). Actually I am replicating a paper("Stock market volatility and macroeconomic fundamentals") in which authors did the same, but I have some troubles in forecasting returns. The conditional mean is the constant, of course, but how can I forecast the levels using the garch? I mean, I know that garch models are used to define confidence intervals for predictions, but the authors showed only the MSE, so I think that they forecasted the levels and compared those with the true ones. I would try a Montecarlo simulation with the white noise part of my garch, but actually I don't think this could be the correct answer to my problem. Can someone kindly give me a hint? Thanks in advance

Answer 1

A GARCH($r$,$s$) model specifies the whole conditional distribution of a variable. So you have \begin{aligned} y_t &= \mu_t + u_t, \\ \mu_t &= \dots \text{(e.g. a constant or an ARMA equation without the term $u_t$)}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dotsc + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dotsc + \beta_r \sigma_{t-r}^2, \\ \varepsilon_t &\sim i.i.d(0,1), \\ \end{aligned} The levels are given by $\mu_t$. If it is a constant, then the forecast is a constant. If it is ARMA, then the forecast is based on the ARMA model. As simple as that.