Moving-average model

The notation MA(q) refers to the moving average model of order q: : X_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \cdots + \theta_q \varepsilon_{t-q} = \mu + \sum_{i=1}^q \theta_i \varepsilon_{t-i} + \varepsilon_{t}, where \mu is the mean of the series, the \theta_1,...,\theta_q are the coefficients of the model and \varepsilon_t, \varepsilon_{t-1},..., \varepsilon_{t-q} are the 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 :X_t = \mu + (1 + \theta_1 B + \cdots + \theta_q B^q)\varepsilon_t. 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. ==Interpretation==

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, \varepsilon _{t-1} appears directly on the right side of the equation for X_t. In contrast, in an AR model \varepsilon _{t-1} does not appear on the right side of the X_t equation, but it does appear on the right side of the X_{t-1} equation, and X_{t-1} appears on the right side of the X_t equation, giving only an indirect effect of \varepsilon_{t-1} on X_t. Second, in the MA model a shock affects X values only for the current period and q periods into the future; in contrast, in the AR model a shock affects X values infinitely far into the future, because \varepsilon _t affects X_t, which affects X_{t+1}, which affects X_{t+2}, and so on forever (see Impulse response). ==Fitting the model==

Fitting a moving-average model is generally more complicated than fitting an autoregressive model. This is because the lagged error terms are not observable. This means that iterative non-linear fitting procedures need to be used in place of linear least squares. Moving average models are linear combinations of past white noise terms, while autoregressive models are linear combinations of past time series values. ARMA models are more complicated than pure AR and MA models, as they combine both autoregressive and moving average components. ==See also==