Suppose we have a stationary time series $x_{1}, x_{2}, ..., x_{T}$. Goal is to forecast up to $T+h$, i.e., forecast $x_{T+1}, x_{T+2}, ..., x_{T+h}$.
Forecasting methodology:
- Using econometric techniques one can try to fit a model, which describes the data generating process of $\{x\}_{t}$ (e.g., ARIMA or other model).
- Then, having estimated parameters of the model one can simulate using forward Monte Carlo technique many paths up to $T+h$.
- Using simulated paths, one can construct an empirical PDF for each point in time in the future, i.e., $f_{T+1}(x), f_{T+2}(x), ..., f_{T+h}(x)$.
- As a forecast for $x_{T+1}, x_{T+2}, ..., x_{T+h}$, take mean of $f_{T+1}(x), f_{T+2}(x), ..., f_{T+h}(x)$, respectively.
Question: Is this kind of method acceptable or widely used for stationary process forecasting? What are the main problems and assumptions of this method? Also, I assume that forecasts will be the same as the current value, i.e., $E(x_{T+i}|x_T)=x_{t}$ for any $i$. Is this assumption correct?