An MA model of order q uses q past disturbances rather than lagged values of the time series in a regression-like model, as follows:
Since we do not observe the white-noise disturbance values, εt, MA(q) is not a regression model like the ones we have seen so far. Rather than using least squares, MA(q) models are estimated using maximum likelihood (MLE), alternatively initializing or estimating the disturbances at the beginning of the series and then recursively and iteratively computing the remainder.
The MA(q) model gets its name from representing each value of yt as a weighted moving average of the past q innovations. In other words, current estimates represent a correction relative to past errors made by the model. The use of moving averages in MA(q) models differs from that of exponential smoothing or the estimation of seasonal time series components because an MA(q) model aims to forecast future values as opposed to de-noising or estimating the trend cycle of past values.
MA(q) processes are always stationary because they are the weighted sum of white noise variables that are themselves stationary.