# Moving-average Models

In time series analysis, a moving-average model is a model that predicts the value of a signal given a linear combination of the lagged errors of the model and the mean of the series.

Mathematically

\[ \displaystyle X_{t}=\mu +\varepsilon _{t}+\theta _{1}\varepsilon _{t-1}+\cdots +\theta _{q}\varepsilon _{t-q} \]

or

\[ \displaystyle X_{t}=\mu +(1+\theta _{1}B+\cdots +\theta _{q}B^{q})\varepsilon _{t} \]

The moving-average is thus a linear regression of the current value of the series against current and previous observed errors. The errors are assumed to be mutually independent and to originate from the same, normal distribution, centred at 0 with a standard deviation of 1.

There can be various “orders” of a moving-average model, denoted by the letter \(q\) in the equations above. We typically refer to a moving-average model with the \(MA(q)\) notation.

Just how in autoregressive models we make use of PACF analysis for determining the order of the model, we can do the same in MA models by using ACF analysis.

Note, the moving-average model can only be used on stationary time series