ARMA
An ARMA model describes a stationary time series using two sources of short-memory dependence: previous values of the series and previous innovations. It is the stationary core inside ARIMA, but without differencing. If a level, trend, or seasonal pattern remains, the series is not ready for ARMA; first revisit stationarity, decomposition, or a seasonal model.
An ARMA process is usually written as
The AR side says the current value partly persists from its own past. The MA side says recent shocks still echo through the series because earlier forecast errors were not fully absorbed at the moment they arrived. In lag-polynomial notation,
The assumptions behind that compact equation matter. The AR polynomial must imply a stable process rather than an exploding one; the MA polynomial should be invertible so the same autocorrelation pattern does not have multiple equivalent parameterizations. Under those conditions, ARMA captures autocorrelation that fades with lag rather than trend that permanently moves the level.
The diagnostic shape is the reason ARMA is often taught alongside autocorrelation and partial autocorrelation. A pure autoregressive model tends to have a PACF cutoff and a gradually decaying ACF. A pure moving-average model tends to have the opposite. Mixed ARMA models usually show less tidy patterns, so order choice is confirmed through residual checks and out-of-sample validation, not just plot reading.
ARMA is most useful for residual processes: sensor deviations after removing a daily cycle, financial returns after demeaning, or one-step forecast errors from a simpler structural model. If the original level series needs differencing, use ARIMA; if the dependence appears at seasonal lags, use SARIMA.
Connections
ARMA combines autoregressive models and moving-average models under a stationarity assumption. Its residuals and candidate orders are read through autocorrelation and partial autocorrelation, while its nonstationary extension is ARIMA.
References
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