What does MA(q) model forecast? Future $X_t$ or future $\epsilon_t$

Question

In time series, Moving Average model $MA(q)$ is defined by $$X_t = \mu + \epsilon_t + \theta_1 \epsilon_{t-1} + \theta_2\epsilon_{t-2} + ... + \theta_q \epsilon_q$$ where $\mu$ is the mean of the process $\{X_t\}_t$ and $\epsilon_t$ is a white noise, that is, uncorrelated with zero mean and fixed variance for all t.$

For simplicity, assume that $\mu = 0.$ From equation above, $MA(q)$ model forecasts future using current and past random shocks (or stochastic terms). Does it mean that we can apply anything to forecasting in $MA(q)$ as long as it is white noise?

I get confused because it seems like the model is not forecasting future $X_t$ but rather forecasting future random shock $\epsilon_{t+1}.$

Answer 1

Why do you get the impression that the future random shocks $\epsilon_{t+1}$ is predicted? That would not make any sense, since in the model the series of random shocks $\dotsc,\epsilon_t, \epsilon_{t+1},\dotsc$ is supposed to represent white noise, which by definition can't be (usefully) forecast, since it is independent of the past.

So, what is forecast in this model is the future observation $X_{t+1}$. For more information about MA models see Moving-average model error terms.