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Autoregression and Moving Average (ARMA) Models

Autoregression and Moving Average (ARMA) Models:

The autoregression and moving average (ARMA) models are used in time series analysis to describe stationary time series . These models represent time series that are generated by passing white noise through a recursive and through a nonrecursive linear filter , consecutively . In other words, the ARMA model is a combination of an autoregressive (AR) model and a moving average (MA) model .

The order of the ARMA model in discrete time Math image is described by two integers Math image , that are the orders of the AR- and MA- parts, respectively. The general expression for an ARMA-process Math image is the following:


  • Math image is the order of the AR-part of the ARMA model;
  • Math image are the coefficients of the AR-part of the model (of the recursive linear filter);
  • Math image is the order of the MA-part of the ARMA model;
  • Math image are the coefficients of the MA-part of the model (of the non-recursive linear filter);
  • Math image are elements of the (input) white noise;
  • Math image are output uncorrelated errors.

The ARMA model, for example, is used to construct the ARIMA model of nonstationary time series .

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