Moving-average model

In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariate time series. The moving-average model specifies that the output variable is cross-correlated with a non-identical to itself random-variable. Together with the autoregressive (AR) model, the moving-average model is a special case and key component of the more general ARMA and ARIMA models of time series, which have a more complicated stochastic structure. Contrary to the AR model, the finite MA model is always stationary. The moving-average model should not be confused with the moving average, a distinct concept despite some similarities. The notation MA(q) refers to the moving average model of order q: where is the mean of the series, the are the parameters of the model and the are white noise error terms. The value of q is called the order of the MA model. This can be equivalently written in terms of the backshift operator B as Thus, a moving-average model is conceptually a linear regression of the current value of the series against current and previous (observed) white noise error terms or random shocks. The random shocks at each point are assumed to be mutually independent and to come from the same distribution, typically a normal distribution, with location at zero and constant scale. The moving-average model is essentially a finite impulse response filter applied to white noise, with some additional interpretation placed on it. The role of the random shocks in the MA model differs from their role in the autoregressive (AR) model in two ways. First, they are propagated to future values of the time series directly: for example, appears directly on the right side of the equation for . In contrast, in an AR model does not appear on the right side of the equation, but it does appear on the right side of the equation, and appears on the right side of the equation, giving only an indirect effect of on .