What is MA(q) model input in real world? I understand the AR(p) model: its input is the time series being modelled. I'm completely stuck when reading about the MA(q) model: its input is innovation or random shock as it's often formulated.
The problem is I can't imagine how to get an innovation component having no model of the (perfect) time series already (i.e. I think $\varepsilon=X_{\rm observed}-X_{\rm perfect}$, and that's probably wrong). Moreover, if we can get this innovation component in-sample, how can we get it when doing a long-term forecast (model error term as a separate additive time series component)?
 A: When trying to get an intuitive real world picture of MA or AR (or ARMA or ARIMA if you are extending it) I often find it useful to think of carry over effects, that is something happening in one period carries over into the next. 
Here's an example: say you are modelling newspaper sales. The noise (random error) in such a model could sensibly incorporate the relatively short lived effect of newspaper headlines while the rest of the model deals with more stable things like trend and seasonality (now I'm assuming an ARIMA model but if you want a pure MA model imagine no trend or seasonality for the paper). Although the newspaper headline effect is modelled as error we might decide that this effect does indeed carry over into the next few days (a good story brings in readers who then fade away again). This would invite the inclusion of an MA term in the model - the carryover of the effect of the previous error term into the current time period.
You can think in the same way about the AR term only what is carried over here is part of the effect of the whole of the previous days sales.
Hope that helps
A: When the unobserved error terms are autocorrelated, there are at least 4 possible strategies since you can't just add the errors into your model:


*

*Use OLS with a corrected variance-covariance matrix (such as Newey-West)

*Transformation of the model

*Feasible Generalized Least Squares

*Instrumental Variables


(2) is probably most common. OLS and FGLS are appropriate for non-scalar residual variance matrices. IV is good when you have a regressor correlated with the error term. Transformations can be useful for both.
Prais-Winsten and Conchrane-Orcutt are common examples of (3) for first-order autocorrelation. These links will illustrate the mechanics nicely.
This post includes some real world examples. In the coupon example, you might imagine adding them as regressors if you could get the data. In the other examples, that makes less sense and (1)-(4) provide a feasible alternative. 
