Autoregressive moving average model
, ARMA (p,q), comprising AR (p) and MA(q) components (Cryer, 1986; Sevuktekin and Nargelecekenler, 2010; Enders, 2010) can be written as in Eq.4.
Autoregressive Moving Average model
abbreviated as ARMA(p,q) model developed by Box and Jenkins  is defined by the combined autoregressive and the Moving Average model.
Thus, this combination characterizes the model defined by the literature as Autoregressive Moving Average Model
ARIMA (autoregressive integrated moving average model) was introduced by Box and Jenkins in the '70, and represents a generalization of ARMA (autoregressive moving average model
Several time series models have been used, such as autoregressive model (AR) or autoregressive moving average model
We find use of random walk or RW (like in Atkeson and Ohanian (2001)); autoregressive moving average model
of integration of order 1 or ARIMA (like in Narayan and Cicarelli (1982), Claus and Claus (2002), Benkovskis (2008) and Adebiyi et al (2014)); and autoregressive model of order 1 or (AR(1) (like in Faust and Wright (2013)) as benchmark models.
* Autoregressive Moving Average Model
of order (p, q), denoted by ARMA (p, q)
Consider the two-input single-output system, described by the following controlled autoregressive moving average model
, depicted in Figure 1:
Autoregressive Moving Average Model
. Autoregressive moving average model
was introduced by Box and Jenkins , for times series prediction, which was actually inspired by the early work of Yule  and Wold .
He describes several nonlinear system identification methods, but focuses particularly on nonlinear autoregressive moving average model
with exogenous inputs (NARMAX) methods.
The autoregressive moving average model
with order p and q (ARMA(p, q)) is given as below:
Some of these procedures, which have their methodological based in PA, help tackle the specification of a particular kind of vector autoregressive moving average model
, such as the transfer function (TF) model [Box and Jenkins, 1976].