VARMAX Procedure
VARMAX Model
The vector autoregressive moving-average model with exogenous variables is called the VARMAX(p,q,s) model. The form of the model can be written as
where the output variables of interest, 
, can be influenced by other input variables, 
, which are determined outside of the system of interest. The variables 
are referred to as dependent, response, or endogenous variables, and the variables 
are referred to as independent, input, predictor, regressor, or exogenous variables. The unobserved noise variables, 
, are a vector white noise process.
The VARMAX(p,q,s) model can be written
where
are matrix polynomials in B in the backshift operator, such that 
, the 
and 
are 
matrices, and the 
are 
matrices.
The following assumptions are made:
, 
, which is positive-definite, and 
for 
. For stationarity and invertibility of the VARMAX process, the roots of
and 
are outside the unit circle. The exogenous (independent) variables
are not correlated with residuals 
, 
. The exogenous variables can be stochastic or nonstochastic. When the exogenous variables are stochastic and their future values are unknown, forecasts of these future values are needed to forecast the future values of the endogenous (dependent) variables. On occasion, future values of the exogenous variables can be assumed to be known because they are deterministic variables. The VARMAX procedure assumes that the exogenous variables are nonstochastic if future values are available in the input data set. Otherwise, the exogenous variables are assumed to be stochastic and their future values are forecasted by assuming that they follow the VARMA(p,q) model, prior to forecasting the endogenous variables, where p and q are the same as in the VARMAX(p,q,s) model.
State Space Representation
Another representation of the VARMAX(p,q,s) model is in the form of a state variable or a state space model, which consists of a state equation
and an observation equation
where
and
On the other hand, it is assumed that follows a VARMA(p,q) model
The model can also be expressed as
where 
and 
are matrix polynomials in B, and the 
and 
are 
matrices. Without loss of generality, the AR and MA orders can be taken to be the same as the VARMAX(p,q,s) model, and 
and 
are independent white noise processes.
Under suitable conditions such as stationarity, is represented by an infinite order moving-average process
where 
.
The optimal minimum mean squared error (minimum MSE) i-step-ahead forecast of 
is
For 
,
The VARMAX(p,q,s) model has an absolutely convergent representation as
or
where 
, 
, and 
.
The optimal (minimum MSE) i-step-ahead forecast of 
is
for 
with 
. For ,
where 
.
Define 
. For 
with , you obtain
From the preceding relations, a state equation is
and an observation equation is
where
and
Note that the matrix K and the input vector 
are defined only when 
.