MA Intuition

MA is a bit unusual because the predictors, i.e., the lagged errors \(\{\varepsilon_{t-1},\dots,\varepsilon_{t-q}\}\) are unobservable, so you cannot fit it by ordinary least squares as in a standard regression. Instead, specialized estimation (e.g. maximum likelihood or method of moments) is required.

Intuition: Imagine you are modeling daily newspaper sales. Suppose yesterday’s headline was a sensational tragedy (an unexpected “error”). That shock brings in extra readers today, and perhaps a bit tomorrow, before its effect fades. In MA terms, \(\varepsilon_{t-1}\) (yesterday’s surprise/error) directly influences \(y_t\) (today’s sales), and may (shortly) in the future. Thus, MA models capture how past shocks/errors “carry over” for a limited number of lags.