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 
is described by two integers 
, that are the orders of the AR- and MA- parts, respectively. The general expression for an ARMA-process 
is the following:
 |
where
is the order of the AR-part of the ARMA model;
are the coefficients of the AR-part of the model (of the recursive linear filter);
is the order of the MA-part of the ARMA model;
are the coefficients of the MA-part of the model (of the non-recursive linear filter);
are elements of the (input) white noise;
are output uncorrelated errors.
is the order of the AR-part of the ARMA model;
are the coefficients of the AR-part of the model (of the 
is the order of the MA-part of the ARMA model;
are the coefficients of the MA-part of the model (of the non-
are elements of the (input) white noise;
are output uncorrelated errors.The ARMA model, for example, is used to construct the ARIMA model of nonstationary time series .