Moving-average model

The notation MA(q) refers to the moving average model of order q:

${\displaystyle X_{t}=\mu +\varepsilon _{t}+\theta _{1}\varepsilon _{t-1}+\cdots +\theta _{q}\varepsilon _{t-q}=\mu +\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}+\varepsilon _{t},}$

where ${\displaystyle \mu }$ is the mean of the series, the ${\displaystyle \theta _{1},...,\theta _{q}}$ are the parameters of the model and the ${\displaystyle \varepsilon _{t},\varepsilon _{t-1},...,\varepsilon _{t-q}}$ 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[4]

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

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.

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

A moving-average model can be fit in the context of time-series analysis by smoothing the time series curve by computing the average of all data points in a fixed-length window.[5] This technique is known as Moving Average Smoothing and can be used for data preparation, feature engineering, and forecasting.[6] Autoregressive Integrated Moving Average (ARIMA) models are an alternative to segmented regression that can also be used for fitting a moving-average model.[7]

Fitting a moving-average model is generally more complicated than fitting an autoregressive model.[8] 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.[9] ARMA models are more complicated than pure AR and MA models, as they combine both autoregressive and moving average components.[8]

The autocorrelation function (ACF) of an MA(q) process is zero at lag q + 1 and greater. Therefore, we determine the appropriate maximum lag for the estimation by examining the sample autocorrelation function to see where it becomes insignificantly different from zero for all lags beyond a certain lag, which is designated as the maximum lag q.

Sometimes the ACF and partial autocorrelation function (PACF) will suggest that an MA model would be a better model choice and sometimes both AR and MA terms should be used in the same model (see Box–Jenkins method).

• Autoregressive–moving-average model
• Autoregressive integrated moving average
• Autoregressive model
• Finite impulse response
• Infinite impulse response

• Enders, Walter (2004). "Stationary Time-Series Models". Applied Econometric Time Series (Second ed.). New York: Wiley. pp. 48–107. ISBN 0-471-45173-8.

• Common approaches to univariate time series

 This article incorporates public domain material from the National Institute of Standards and Technology.